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# What is the probability that an integer between 0 and 9,999 has exactly one 8 and one 9?
I Googled this question and found some answers but they were all different from each other, so I don't know which one is correct.
Question: What is the probability that an integer between 0 and 9,999 has exactly one 8 and one 9?
I tried by splitting it into 3 cases.
Case #1 (2 digits): 98 or 89, so only 2!=2 possible ways of arranging 2 digits.
Case #2 (3 digits): One possible outcome is 98y where y can be integers from 0 to 7 (so there's 8 possible values for y). 3!=6 ways of arranging it. So there's 6*8-2 = 46 possible 3 digit numbers. I subtracted 2 from it because 098 and 089 are not 3 digit numbers.
Case #3 (4 digits): A possible outcome is 98xy where x and y can both take on 8 possible values ranging from 0 to 7. 4!/2! = 12 ways of arranging it because if x=y, 98xy is the same as 98yx. So we have: 12*8*8-48 = 720 possible 4 digit numbers. 48 is the result of 46 + 2 from case #2 because we have to subtract outcomes where the first two digits are 0 or the first digit is a 0 eg: 0098.
Therefore, the total is 2 + 46 + 720 = 768 numbers with exactly 8 and 9 in it. The probability is 768/10000. | {
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• Note that if you allow the four digit numbers to have leading zeroes (i.e. allow numbers like $0098$), then that takes care of two- and three-digit numbers automatically, without having to treat those as a special case, and without having to treat $0$ as a special digit. – Arthur Jan 22 '16 at 21:20
• Your result is correct. It is a much harder approach than the answers, but you were careful to get them all and once each. – Ross Millikan Jan 22 '16 at 21:29
• leading zeros are allowed because associating strictly 4 digit numbers with leading zeros allowed with the corresponding 1, 2, 3 or 4 digit numbers without leading zeros is a 1-1 association. e.g. 0071 ~ 71 and nothing else. ... or another way of putting it 0071 and 71 are simply two different ways of writing the same thing. And as the two answers show, using leading zeros makes the calculations much easier. – fleablood Jan 22 '16 at 21:45
Let us ask the related question of how many 4-digit strings have exactly one eight and one nine. (Strings are allowed to have leading zeroes, whereas numbers are not) Note that the 4-digit strings are in direct bijection with the integers from $0$ to $9999$.
• Pick the location of the nine ($4$ choices)
• Pick the location of the eight (It can't be where the nine is, so $3$ choices)
• For each remaining position, pick a digit other than eight or nine ($8$ choices each time for a total of $8\cdot 8$ choices)
Thus, the number of four-digit strings with exactly one eight and one nine is $4\cdot 3\cdot 8\cdot 8$
Since there are $10^4$ different four-digit strings regardless, the probability is then $\dfrac{4\cdot 3\cdot 8\cdot 8}{10^4} = \dfrac{768}{10000} = .0768$
If you make all your numbers four digits by allowing leading zeros it is easier. You have four places to put the $9$, then three left to put the $8$, then $8$ choices for each of the other two places. $4 \cdot 3 \cdot 8^2=768$ good numbers out of $10000$, so the probability is $0.0768$ | {
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Power Series Calculator Differential Equations | {
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An important application of power series in the field of engineering is spectrum analysis. Then solving for ∆Φ gives 0. Theoretical considerations and convergence of the method for these systems are discussed. The second part of this course introduces series, especially the power series. Covers material on integration methods (trig, partial fractions, etc. Laplace Transform of the Dirac Delta Function using the TiNspire Calculator. This script may help the Calculus (II or III) student with the Infinite Series chapter, and it may also help the Differential Equations student with Series Solutions. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Derivatives Derivative Applications Limits Integrals Integral Applications Series ODE Laplace Transform Taylor/Maclaurin Series Fourier Series. Prev Up Next. ), probability functions, Taylor polynomials/series approximations, power series’, differential equations (linear and separable), partial derivatives, multivariable functions (and their real-world applications), and double integrals in. Let's nd a solution as a= 0. We again use Maple to find the power series solutions as well. The Equation for the Quantum Harmonic Oscillator is a second order differential equation that can be solved using a power series. Tìm kiếm power series solution of differential equations calculator , power series solution of differential equations calculator tại 123doc - Thư viện trực tuyến hàng đầu Việt Nam. Post date: 15 Feb 2011. Yes, y(x) is the general solution of the differential equation represented as a power series. Motivation: Following this discussion about using asymptotic expansions (i. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. James Stewart's Calculus, Metric series is the top-seller in the world because of its | {
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when exchange rate misalignment suppresses capital inflows. Each Differential Equations problem is tagged down to the core, underlying concept that is being tested. Differential Equations: Techniques, Theory, and Applications is designed for a modern first course in differential equations either one or two semesters in length. Xavier Sigaud, 150 CEP 22290-180, Rio de Janeiro, RJ, Brazil. I for jx x 0j<ˆand diverges jx x 0j>ˆ. Approximate solutions of first-order differential equations using Euler and/or Runge-Kutta methods. Houston Math Prep 245,356 views. An important application of power series in the field of engineering is spectrum analysis. The recruit becomes a reproductive adult in the next time step, and begins modifying the abundance of each microbial taxon through time according to the following differential equation:. Although the method may be applied to first order equations, our discussion will center on second order equations. 1126 CHAPTER 15 Differential Equations In Example 1, the differential equation could be solved easily without using a series. This gives a recurrence formula for the coefficients. 2 Power Series, Analytic Functions, and the Taylor Series Method 431 8. Question: In this exercise we consider the second order linear equation {eq}y'' + 4y = 0 {/eq}. On our site OnSolver. Laplace transforms. 7MB) To complete the reading assignments, see the Supplementary Notes in the Study Materials section. Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Induction Pre Calculus Equations Inequalities System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & Comp. Added Aug 1, 2010 by Hildur in Mathematics. EXAMPLE 1 Use power series to solve the equation. Well in order for a series solution to a differential equation to exist at a particular x | {
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equation. Well in order for a series solution to a differential equation to exist at a particular x it will need to be convergent at that x. Differential Equation Calculator. • Proceeding just as for series but now in voltage (1) Using KCL to write the equations: 0 0 1 vdt I R L v dt di C t + + ∫ = (2) Want full differential equation • Differentiating with respect to time 0 1 1 2 2 + + v = dt L dv R d v C (3) This is the differential equation of second order • Second order equations involve 2nd order derivatives. Calculus is the mathematics of change, and rates of change are expressed by derivatives. The term "ordinary" is used in contrast with the term. Let’s consider the equation: 2 ′′ + + ′ x y x x y y − = 2 7 ( 1) 3 0 (1). 3 Systems of ODEs. Use MathJax to format equations. We propose a computational method to determine when a solution modulo a certain power of the independent variable of a given algebraic differential equation (AODE) can be extended to a formal power series solution. 2 - Series and Sequences of Partial Sums; Lesson 21. Power Series Ordinary Differential Equations Esteban Arcaute1 We then need partial fractions to calculate Z dy N(y) = Z Q(y) P(y) dy. For example, diff(y,x) == y represents the equation dy/dx=y. A Differential Equation is a n equation with a function and one or more of its derivatives:. SOLUTION We assume there is a solution of the form. If the function is of only one variable, we call the equation an ordinary differential equation (ODE). So, why are we worried about the convergence of power series? Well in order for a series solution to a differential equation to exist at a particular x it will need to be convergent at that x. This might introduce extra solutions. 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 50 49 48 47 46 45 44 43 42 41. In following section, 2. The number ρ is at least 0, as taking x = x0 gives P 0 which is clearly converging to 0; On the other | {
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number ρ is at least 0, as taking x = x0 gives P 0 which is clearly converging to 0; On the other hand, when the power series is convergent for all x, we say its radius of convergence is infinity,. Sturm-Liouville problems. The differential equation in Example 2 cannot be solved by any of the methods discussed in previous sections. Use the power series method to find 2 linearly independent series solutions to differential equation (x^2 - 2x + 2)y" + 2(x-1)y' - 2y = 0 about the ordinary point xo=1. Sage Quickstart for Differential Equations¶. Pourhabib Yekta1, A. Browse other questions tagged sequences-and-series ordinary-differential-equations power-series or ask your own question. The differential equation can be writ-ten in. After a promising start to his mathematical career, Nash began to suffer from schizophrenia around his 30th year, an illness from which he has. An older book that has a lot of nice material on power series and other numerical methods for ODE's is Einar Hille's Lectures On Ordinary Differential Equations. Let’s consider the equation: 2 ′′ + + ′ x y x x y y − = 2 7 ( 1) 3 0 (1). With the exception of special types, such as the Cauchy equations, these will generally require the use of the power series techniques for a solution. The Bessel differential equation has the form x 2 y+xy'+(x 2-n 2)y=0. Power Series Method for Nonlinear Partial Differential Equations Power series is an old technique for solving linear ordinary differential equations [7,20]. Approximate solutions of first-order differential equations using Euler and/or Runge-Kutta methods. The general Airy differential equation is given by :$D^2y \pm m^2 x y = 0$or equivalently$y’’ \pm m^2 x y = 0$The differential equation in the question is a form of the Airy differential equation with the minus sign preceding $m^2$. Chapter 7 Power series methods 7. 1 in [BD] Many functions can be written in terms of a power series X1 k=0 a k(x x 0)k: If we assume that a solution of a di erential | {
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written in terms of a power series X1 k=0 a k(x x 0)k: If we assume that a solution of a di erential equation is written as a power series, then perhaps we can use a method reminiscent of undetermined coe cients. In this section we learn how to extend series solutions to a class of differential equations that appear at first glance to diverge in our region of interest. Solution of linear equations by power series Def. Use the power series method to find 2 linearly independent series solutions to differential equation (x^2 - 2x + 2)y" + 2(x-1)y' - 2y = 0 about the ordinary point xo=1. Given three points, A, , , B, , , and C, , : a Specify the vector A extending from the origin to the point A. An infinite series of this type is called a power series. An excellent article in the American Journal of Physics, by Fairen, Lopez, and Conde develops power series approximations for various systems of nonlinear differential equations. With the exception of special types, such as the Cauchy equations, these will generally require the use of the power series techniques for a solution. 8 Nonlinear systems. Linear and separable first order differential equations. math 230 psu reddit, Review of calculus, properties of real numbers, infinite series, uniform convergence, power series. Finding coefficients in a power series expansion of a rational function. shirin setayesh 55,974 views. 79: conditions calculate Cauchy-Euler equation portrait population pounds power series predictions recurrence. 3 - Recursively Defined Sequences. This method aims to find power series for the solution functions to a differential equation. Equations relating the partial derivatives (See: Vector calculus ) of a function of several variables are called partial differential equations (PDEs). Example $$\PageIndex{2}$$ Find the the first three nonzero terms of two linearly independent solutions to $$xy'' + 2y = 0$$. Equate coefficients of like powers of $$x$$ to determine values for the coefficients $$a_n$$ | {
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= 0$$. Equate coefficients of like powers of $$x$$ to determine values for the coefficients $$a_n$$ in the power series. If initial conditions are given, determine the particular solution. Home PDF Paperback Index PrevUp Next. Power series solution is a method to solve the differential equations. • Use power series to find solutions to higher order linear differential equation with nonconstant coefficients at any regular singular point; and • Use Laplace transforms to solve initial value problems. Comprehensive & Detailed COMPLETE note package for the course Math 128 (Calculus II). This RPS method gives approximate solutions in convergent series formula with surely computable components. The differential equation can be writ-ten in. Shifting the Index for Power Series - Duration: 14:49. Differential Equations for Engineers. ’s need to be. Research Article Power Series Extender Method for the Solution of Nonlinear Differential Equations HectorVazquez-Leal 1 andArturoSarmiento-Reyes 2 Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. If the function is of only one variable, we call the equation an ordinary differential equation (ODE). An important application of power series in the field of engineering is spectrum analysis. Power series neural network solution for ordinary differential equations with initial conditions Abstract: Differential equations are very common in most academic fields. {image} {image} {image} {image} 3. Now We have two components R and L connected in Series and a voltage source to those components as shown below. For the differential equation show below:y(1) = 1y'(1) = 0a) Write down a general expansion for a power series solution y(x) about the point Xo = 1b)Find a recurrence relationshi between the coefficients of your power series | {
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about the point Xo = 1b)Find a recurrence relationshi between the coefficients of your power series expansion. Notice that 0 is a singular point of this differential equation. Get this from a library! Formal power series and linear systems of meromorphic ordinary differential equations. 3 Power Series Solutions to Linear Differential Equations 442 8. We will only be able to do this if the point x = x0. Since cos x = Σ(n=0 to ∞) (-1)^n x^(2n)/(2n)!, so u can get by potential of heart a million qn and remedy each and every) Non-linear partial differential equation, Homogenous and non-homogeneous. In this video from PatrickJMT we show how to use power series to find a solution of a differential equation. In the equation, represent differentiation by using diff. 3 - Taylor Series. Fourier series with Ti84: Calculate for all different periodic signals the spectrum. y'' − y' = 0. ; Dig deeper into specific steps Our solver does what a calculator won't: breaking down key steps. 2) By insertion of y = n=0 anx n,y = n=1 nanx n 1,y = n=2 n(n 1)anx n 2, Solution of differential equations by the power series method. Illustrative numerical example is included to demonstrate efficiency. equation is given in closed form, has a detailed description. c) Write down the first few terms of your power series (up to the fourth. We have step-by-step solutions for your textbooks written by Bartleby experts!. 1126 CHAPTER 15 Differential Equations In Example 1, the differential equation could be solved easily without using a series. Full curriculum of exercises and videos. A power series solution to a differential equation is a function with infinitely many terms, each term containing a different power of the dependent variable. In following section, 2. Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. AMS30, 151-156 (1971). Underdamped Overdamped Critically | {
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has a radius of convergence equal to zero. AMS30, 151-156 (1971). Underdamped Overdamped Critically Damped. Use MathJax to format equations. Conic Sections Trigonometry. An older book that has a lot of nice material on power series and other numerical methods for ODE's is Einar Hille's Lectures On Ordinary Differential Equations. Since many physical laws and relations appear mathematically in the form of differential equations, such equations are of fundamental importance in engineering mathematics. Derivatives Derivative Applications Limits Integrals Integral Applications Series ODE Laplace Transform Taylor/Maclaurin Series Fourier Series. Learn AP®︎ Calculus BC for free—everything from AP®︎ Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP®︎ test. We’ll assume mechanical boresight (θ = 0º), N = 8, and d = λ/2. Now We have two components R and L connected in Series and a voltage source to those components as shown below. The validity of term‐by‐term differentiation of a power series within its interval of convergence implies that first‐order differential equations may be solved by assuming a solution of the form. The Differential Equations diagnostic test results highlight how you performed on each area of the test. Related Calculators. I am now at this point where I have got: \displaystyle \sum_{n = 0}^&#. It often happens that a differential equation cannot be solved in terms of elementary functions (that is, in closed form in terms of polynomials, rational functions, e x, sin x, cos x, In x, etc. Covers material on integration methods (trig, partial fractions, etc. Video tutorial on Power Series Solutions of Differential Equations - In this video, I show how to use power series to find a solution of a differential equation. This might introduce extra solutions. Differential equation or system of equations, specified as a symbolic equation or a vector of symbolic equations. In dissertation we discuss power series | {
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as a symbolic equation or a vector of symbolic equations. In dissertation we discuss power series characteristics that we use for solving the equations in question. Generalized series expansions involving integer powers and fractional powers in the independent variable have recently been shown to provide solutions to certain linear fractional order differential equations. ; Dig deeper into specific steps Our solver does what a calculator won't: breaking down key steps. Re-index sums as necessary to combine terms and simplify the expression. In our world things change, and describing how they change often ends up as a Differential Equation: an equation with a function and one or more of its derivatives: Introduction to Differential Equations; Differential Equations Solution Guide; Separation of Variables. View and Download PowerPoint Presentations on Solution Differential Equation By Power Series PPT. Even if you don't know how to find a solution to a differential equation, you can always check whether a proposed solution works. Mathematics > Calculus and Analysis > Differential Equations Keywords Calculus, series expansion, Taylor Series, Ordinary Differential Equation, ODE, , Power Series. Order Differential Equations with non matching independent variables (Ex: y'(0)=0, y(1)=0 ) Step by Step - Inverse LaPlace for Partial Fractions and linear numerators. Determining the value of a definite integral on the graphing calculator. Find the singular points (if any) for the following equations. How calculators calculate is by power series. Xavier Sigaud, 150 CEP 22290-180, Rio de Janeiro, RJ, Brazil. Exercises 8. Given a linear differential equation with polynomial coefficients a point x = x 0 is called an ordinary point if b 0 (x 0) 0. Definition 5. Taylor's Series method. 5 The Power Series Method. To leave a comment or report an error, please use the auxiliary blog. Each Differential Equations problem is tagged down to the core, underlying concept that is being tested. | {
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Differential Equations problem is tagged down to the core, underlying concept that is being tested. = 1 for y at x = 1 with step length 0. This Sage quickstart tutorial was developed for the MAA PREP Workshop "Sage: Using Open-Source Mathematics Software with Undergraduates" (funding provided by NSF DUE 0817071). Special Functions The power series method gives solutions of linear ODEs (1) y” + p(x)y’ + q(x)y = 0 with variable coefficients p and q in the form of a power series (with any center x0, e. lol this is a question you'll look back on after doing more math/physics and laugh. I Check the endpoints, jx x 0j= ˆ, separately to nd the INTERVAL of CONVERGENCE. {image} {image} {image} {image} 3. Solutions by separation of variables and expansion in Fourier Series or other appropriate orthogonal sets. 1 - Sequences; Lesson 21. In every upper division physics class you will use a power series. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. Hermite’s differential equation shows up during the solution of the Schrödinger equation for the harmonic oscillator. the menu option in Differential Equations Made Easy from www. AN EXAMPLE. A differential equation, shortly DE, is a relationship between a finite set of functions and its derivatives. Calculus: Difference Equations to Differential Equations ADD. 3 - Recursively Defined Sequences. 2 Power Series, Analytic Functions, and the Taylor Series Method 431 8. In our world things change, and describing how they change often ends up as a Differential Equation: an equation with a function and one or more of its derivatives: Introduction to Differential Equations; Differential Equations Solution Guide; Separation of Variables. 1 Power series Note: 1 or 1. Solution of linear equations by power series Def. Get this from a library! Formal power series and linear | {
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of linear equations by power series Def. Get this from a library! Formal power series and linear systems of meromorphic ordinary differential equations. In dissertation we discuss power series characteristics that we use for solving the equations in question. The general Airy differential equation is given by :$D^2y \pm m^2 x y = 0$or equivalently$y’’ \pm m^2 x y = 0$The differential equation in the question is a form of the Airy differential equation with the minus sign preceding $m^2$. Q3: Find the series solution for the following ordinary differential equation using the Frobenius method: 𝑥 𝑦 + 𝑥 𝑥 + 1 2 𝑦 − 𝑥 + 1 2 𝑦 = 0. 1 of 3 Go to page. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. order Power series solution of differential-algebraic equations in (1. In this lecture, we will study the solution of the second-order linear differential equations in terms of Power Series. Nonlinear Differential Equations Fourier series For a periodicfunction one may write The Fourier series is a "best fit" in the least square sense of data fitting y(t +T) =y(t) ()cos( ) sin( ), a plot of versus is called the power spectrum ∫ +∞ −∞. We also discuss more about initial conditions and how they determine the first two coefficients in the power series solution. Next enter the c value and view the Laplace transform below the entry box. The general idea is as follows: Assume that the solution function has a power series that converges to it. This method aims to find power series for the solution functions to a differential equation. Video # Video Tutorial Title: Remarks: 1: Classification of Differential Equations. We will only be able to do this if the point x = x0. Give the first four non-zero terms of each of the 2 independent solutions, y1. In this thesis, the reader will be made aware of methods for finding power series solutions | {
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y1. In this thesis, the reader will be made aware of methods for finding power series solutions to ordinary differential equations. S = dsolve(eqn) solves the differential equation eqn, where eqn is a symbolic equation. There is a Review Sheet (with Solutions). Module 25 - Parametric. But first: why?. 1 - Power Series; Lesson 22. It's more plug-and-chug and you should do well if you can match up the differential equation to the approach used to solve it. Topic: Differential Equations, Sequences and Series. Module 21 - Sequences and Series; Lesson 21. 2 Linear Ordinary Differential Equations with Constant Coefficients A174 A. In some cases, these power series representations can be used to find solutions to differential equations. The method works analogously for higher order equations as well as for systems. In the equation, represent differentiation by using diff. Convergent Power Series of sech ( x ) and Solutions to Nonlinear Differential Equations Article (PDF Available) in International Journal of Differential Equations 2018(1-2):1-10 · February. However, you can specify its marking a variable, if write, for example, y(t) in the equation, the calculator will automatically recognize that y is a function of the variable t. Taylor Series Calculator - an Introduction. The dsolve function finds a value of C1 that satisfies the condition. Let's nd a solution as a= 0. Hi and welcome back to the differential equations lectures here on www. Below is one of them. Power Series Solution to Non-Linear Partial Differential equations of Mathematical Physics. delay differential equations using the residual power series method (RPSM), which obtains. In the first part of this course, the student learns to solve the most common types of differential equations. Derivative Calculator Integral Calculator Limit Calculator. Fourier series with Ti84: Calculate for all different periodic signals the spectrum. , x0 = 0) (2) y( x) am ( x x0 )m a0 a1 ( x x0 ) a2 ( x x0 )2. What is more, we | {
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signals the spectrum. , x0 = 0) (2) y( x) am ( x x0 )m a0 a1 ( x x0 ) a2 ( x x0 )2. What is more, we present the post-treatment of the power series. Access course-tailored video lessons, exam-like quizzes, mock exams & more for MATH 118 at Waterloo. The dsolve function finds a value of C1 that satisfies the condition. Example: t y″ + 4 y′ = t 2 The standard form is y t t. Differential Equation Calculator. Solved Examples of Differential Equations Sunday, July 9, 2017 Find the first 6 non-zero terms of the power series expansion about x = 0 for a general solution to the given differential equation y'' - x^2y' - xy = 0. Hi and welcome back to the differential equations lectures here on www. Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero. The validity of term‐by‐term differentiation of a power series within its interval of convergence implies that first‐order differential equations may be solved by assuming a solution of the form. Let's study about the order and degree of differential equation. KEYWORDS: Course Materials, Lecture notes, Laboratories, HW Problems SOURCE: Joseph M. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. 2 using Taylor series method of order four. 100, 61111 Ljubljana, Slovenia and Racah Institute of Physics,. 2 Power Series, Analytic Functions, and the Taylor Series Method 431 8. Answer all the questions. This method aims to find power series for the solution functions to a differential equation. Differential equation or system of equations, specified as a symbolic equation or a vector of symbolic equations. Illustrative numerical example is included to demonstrate efficiency. ), probability functions, Taylor polynomials/series approximations, power series’, differential equations (linear and separable), partial derivatives, multivariable functions (and | {
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differential equations (linear and separable), partial derivatives, multivariable functions (and their real-world applications), and double integrals in. {image} {image} {image} {image} 3. Modeling with Differential Equations Solve a problem in the physical sciences (such as a growth or decay problem, a mixture problem, or a Newton’s Law of Cooling problem) whose solution utilizes a first-order linear differential equation. 3 - Taylor Series. The coefficient functions in these series are obtained by recursively iterating a simple integration process, begining with a solution system for $\lambda=0$. This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. By solving such equations, we mean computing a vector F of power series such that (1) holds modulo xN. In the equation, represent differentiation by using diff. 2) for all x implies, by the nth term test for diver-gence, that lim n→∞ n! = 0 (5. 5 Cauchy-Euler (Equidimensional) Equations Revisited 459. Here we have discussed an Ordinary and singular point for linear Second Order Differential Equations, classification of Singular Point and method to solve Differential Equations about an Ordinary Point. Print Book & E-Book. Solving Separable First Order Differential Equations – Ex 1 Homogeneous Second Order Linear Differential Equations Power Series Solutions of Differential Equations. try to explain any differences between the two forms of the solution. 1 - Sequences; Lesson 23. Stefan, Jamova 39, P. EXCHANGE RATE MISALIGNMENT AND CAPITAL INFLOWS: AN ENDOGENOUS THRESHOLD ANALYSIS FOR MALAYSIAABSTRACTThis study presents an attempt to investigate the impact of exchange rate misalignment on capital inflows in Malaysia. Find more Mathematics widgets in Wolfram|Alpha. $y^\prime = 2|x|$ is kind of an artificial example. HERMITE DIFFERENTIAL EQUATION - GENERATING FUNCTIONS Link to: physicspages home page. Why most of Hille's texts-which are all wonderful-are out of print | {
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Link to: physicspages home page. Why most of Hille's texts-which are all wonderful-are out of print mystifies me. Review of Series and Power Series. This Sage quickstart tutorial was developed for the MAA PREP Workshop "Sage: Using Open-Source Mathematics Software with Undergraduates" (funding provided by NSF DUE 0817071). However, you can specify its marking a variable, if write, for example, y(t) in the equation, the calculator will automatically recognize that y is a function of the variable t. Initialization. In a few upper div math classes, like differential equations, real and complex analysis you'll see them. In Introduction to Power Series , we studied how functions can be represented as power series, We also saw that we can find series representations of the derivatives of such functions by differentiating the power series term by term. It is interesting to know whether sympy supports such equations along with usual ones. Why most of Hille's texts-which are all wonderful-are out of print mystifies me. Initialization. Sympy: how to solve algebraic equation in formal. 5 Cauchy-Euler (Equidimensional) Equations Revisited 459. Equations Inequalities System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & Comp. Differential equation. It is shown how to obtain such an initializing system working upwards from equations of lower order. Sage Quickstart for Differential Equations¶. He shared the 1994 Nobel Prize for economics with two other game theorists, Reinhard Selten and John Harsanyi. Order Differential Equations with non matching independent variables (Ex: y'(0)=0, y(1)=0 ) Step by Step - Inverse LaPlace for Partial Fractions and linear numerators. My class, and many other's, continue onto power series solutions of differential equations. Convergent Power Series of sech ( x ) and Solutions to Nonlinear Differential Equations Article (PDF Available) in International Journal | {
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) and Solutions to Nonlinear Differential Equations Article (PDF Available) in International Journal of Differential Equations 2018(1-2):1-10 · February. Power Series Method for Linear Partial Differential Equations of Fractional Order 73 Lemma 2. Practice your math skills and learn step by step with our math solver. Solve y0 +(2x 1)y = 0 with initial conditions y(0) = 2. SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS— SOME WORKED EXAMPLES First example Let’s start with a simple differential equation: ′′− ′+y y y =2 0 (1) We recognize this instantly as a second order homogeneous constant coefficient equation. Finding coefficients in a power series expansion of a rational function. Answers to Solving Ordinary Differential Equations with Power Series Here are the answers to the practice questions I provide throughout this chapter. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. In fact, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power. Power series representations of functions can sometimes be used to find solutions to differential equations. Differential Equations. Mathematical concepts and various techniques are presented in a clear, logical, and concise manner. Learn AP®︎ Calculus BC for free—everything from AP®︎ Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP®︎ test. Initial conditions are also supported. This page covers two areas related to Fourier Series. We will only be able to do this if the point x = x0. If m 1 < 0 The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of. Solving Separable First Order Differential Equations – Ex 1 Homogeneous Second Order Linear Differential Equations Power Series Solutions of Differential Equations. One-variable linear | {
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Linear Differential Equations Power Series Solutions of Differential Equations. One-variable linear equations Calculator; One-variable linear inequalities Calculator; Operations with infinity Calculator; Perfect square trinomial Calculator; Polynomial factorization Calculator; Polynomial long division Calculator; Polynomials Calculator; Power of a product Calculator; Power rule Calculator; Power series Calculator; Powers. Homogeneous Differential Equations Calculator. equation, method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method); Legendre and Bessel functions and their orthogonal properties; Systems of linear first order ordinary differential equations. This RPS method gives approximate solutions in convergent series formula with surely computable components. 5 lecture , §3. 1 First order equations. The Organic. Module 23 - Parametric. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. In this work, we studied that Power Series Method is the standard basic method for solving linear differential equations with variable coefficients. MA 401 Applied Differential Equations II (Wave, heat and Laplace equations. I Check the endpoints, jx x 0j= ˆ, separately to nd the INTERVAL of CONVERGENCE. {image} {image} {image} {image} 3. You can view the Laplace Table that you will be given on the Exam. Practice 2610. Specifically, a precise threshold value is estimated to examine when exchange rate misalignment suppresses capital inflows. You can view the Laplace Table that you will be given on the Exam. This is simply a matter of plugging the proposed value of the dependent variable into both sides of the equation to see whether equality is maintained. The solution diffusion. But first: why?. John Forbes Nash Jr Essay John Forbes Nash Jr. lol this is a question you'll look | {
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But first: why?. John Forbes Nash Jr Essay John Forbes Nash Jr. lol this is a question you'll look back on after doing more math/physics and laugh. Example The differential equation ay00 +by0 +cy = 0 can be solved by seeking exponential solutions with an unknown exponential factor. Power Series; Method of series solutions; 6 The Laplace. After finding the constants a 2 ,a 3 ,a 4 , etc I replaced them in y(x) and factored out the undetermined coefficients a 0 and a 1. 7 Power series methods. using traditional way with pencil and paper. Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran2 Abstract: In this article power series method, as well-known method for. The extension of fractional power series solutions for linear fractional differential equations with variable coefficients is considered. Since cos x = Σ(n=0 to ∞) (-1)^n x^(2n)/(2n)!, so u can get by potential of heart a million qn and remedy each and every) Non-linear partial differential equation, Homogenous and non-homogeneous. SOLVING DIFFERENTIAL EQUATIONS USING POWER SERIES 4 (2) Plug the expression (1) for y(x) into the di erential equation; (3) Manipulate the resulting equation to obtain an equation in which single power series expression (rather that a sum of several power series) is set equal to zero. In this work, we studied that Power Series Method is the standard basic method for solving linear differential equations with variable coefficients. We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. 2, the power series method is used to derive the wave function and the eigenenergies for the quantum harmonic oscillator. THE METHOD OF FROBENIUS We have studied how to solve many differential equations via series solutions. The coe cient functions here are constants, so the power series solution can be computed at any point aand the radius of convergence will be R=1. Houston Math Prep 245,356 views. Conic Sections | {
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point aand the radius of convergence will be R=1. Houston Math Prep 245,356 views. Conic Sections Trigonometry. After finding the constants a 2 ,a 3 ,a 4 , etc I replaced them in y(x) and factored out the undetermined coefficients a 0 and a 1. Stefan, Jamova 39, P. using traditional way with pencil and paper. Together we will learn how to express a combination of power series as a single power series. Applications of Fourier Series to Differential Equations Fourier theory was initially invented to solve certain differential equations. I used the power series method to solve the differential equation y''+y=0 with. So, the convergence of power series is fairly important. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. To pursue these objectives, this study relies on the. James Stewart's Calculus, Metric series is the top-seller in the world because of its problem-solving focus, mathematical precision and accuracy, and outstan. This particular number ρ is called the r adiu s of c onv er ge nc e. Equations Inequalities System of Equations System of Inequalities Polynomials Rationales Coordinate Geometry Complex Numbers Polar/Cartesian Functions Arithmetic & Comp. We will only be able to do this if the point x = x0. Question: Find two power series solutions of the given differential equations about the ordinary point {eq}x = 0 {/eq}. Solve differential equation using "Power Series" Thread starter EmilyL; Start date Jul 23, 2010; Tags differential differential equations equation power series solve; Home. ; Dig deeper into specific steps Our solver does what a calculator won't: breaking down key steps. HAFTEL Code 6651, Naval Research Laboratory, Washington, DC 20735-5345 R. Browse other questions tagged ordinary-differential-equations power-series or ask your own question. Motivation: Following this discussion about using asymptotic expansions (i. ODEs Summer08 Esteban Arcaute Introduction First Order ODEs Separation | {
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asymptotic expansions (i. ODEs Summer08 Esteban Arcaute Introduction First Order ODEs Separation of Power Series Exact Equation End Thus, if the equation is exact, we have f(x,y) = c. Module 25 - Parametric. 307) than what Sal got by raising both sides to the power of e: 2. , in the form. KEYWORDS: Course Materials Calculus for Biology I ADD. Use power series to solve the differential equation. The chapter discusses a method by which the coefficients in the power series expansions of the solutions can be calculated. Those I have learnt in lecture and online are mostly with only one part of summation or two parts with two distinctive roots and two constants. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Linear and separable first order differential equations. Xavier Sigaud, 150 CEP 22290-180, Rio de Janeiro, RJ, Brazil. AN EXAMPLE. Up to 25 % of the generating costs relate to mainte- nance. Mathematics > Calculus and Analysis > Differential Equations Keywords Calculus, series expansion, Taylor Series, Ordinary Differential Equation, ODE, , Power Series. 4 Fourier series and PDEs. Substituting y = ert into the equation gives a solution if the quadratic equation ar2 +br+c = 0 holds. compare the series solutions with the solutions of the differential equation obtained using the method of section 4. Welcome! This is one of over 2,200 courses on OCW. Depending upon the domain of the functions involved we have ordinary differ-ential equations, or shortly ODE, when only one variable appears (as in equations (1. Let's consider the power series solution of the Hermite differential equation: ${\displaystyle u''-2xu'=-2 n u}$ ${\displaystyle u''-2xu'+2 n u =0 \qquad (1)}$ The solutions to the Hermite differential equation ca be expresse. EXAMPLE2 Power Series Solution Use a power series to solve the differential equation Solution Assume that is a. 5 lecture , §3. Solving the Systems of Differential Equations | {
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equation Solution Assume that is a. 5 lecture , §3. Solving the Systems of Differential Equations by a Power Series Method A. Conic Sections Trigonometry. An infinite series of this type is called a power series. form a fundamental system of solutions for Airy's Differential Equation. Shifting the Index for Power Series - Duration: 14:49. Textbook solution for Calculus: Early Transcendentals 8th Edition James Stewart Chapter 17. Such an expression is nevertheless an entirely valid solution, and in fact, many specific power series that arise from solving particular differential equations have been extensively studied and hold prominent places in mathematics and physics. ; Dig deeper into specific steps Our solver does what a calculator won't: breaking down key steps. 3 - Second-Order Differential Equations. We will solve this using power series technique. Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [26-34]). Worked example: exponential solution to differential equation. Solving Differential Equations with Power Series - Duration: 18:29. Do the differential equation solvers - Support ordinary differential equations; systems of differential equations, and boundary value problems both at the command line and in solve blocks that use natural notation to specify the DiffEQs and constraints. 4 The Power Series Method, Part I A187 A. Finding coefficients in a power series expansion of a rational function. For example, here's a differential equation […]. Let's consider the power series solution of the Hermite differential equation: ${\displaystyle u''-2xu'=-2 n u}$ ${\displaystyle u''-2xu'+2 n u =0 \qquad (1)}$ The solutions to the Hermite differential equation ca be expresse. In the cases where series cannot be reduced to a closed form expression an approximate answer could be obtained using definite integral calculator. lol this is a question you'll | {
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answer could be obtained using definite integral calculator. lol this is a question you'll look back on after doing more math/physics and laugh. For example, diff(y,x) == y represents the equation dy/dx=y. Find the Taylor series expansion of any function around a point using this online calculator. We will only be able to do this if the point x = x0. 3 - Recursively Defined Sequences. And find the power series solutions of a linear first-order differential equations whose solutions can not be written in terms of familiar functions such as polynomials, exponential or trigonometric functions, as SOS Math so nicely states. The Bessel differential equation has the form x 2 y+xy'+(x 2-n 2)y=0. Around the Point a = (default a = 0) Maximum Power of the Expansion: How to Input. 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 50 49 48 47 46 45 44 43 42 41. Definition 5. ’s need to be. Introduction to Differential Equations. Write y' as dy/dx and the answer follows relatively - dy/dx + 3(a million+ x^2)y = 0 => dy/dx = -3(a million+x^2)y => dy/y = -3(a million+x^2)dx Now integrating the two components supplies ln(y) = -3x - x^3 + C (C is unknown integration consistent) => y = ok exp(-3x - x^3) the place ok = exp(C), and can be solved utilising extra suitable education approximately y (as an occasion the fee of y. In advanced treatments of calculus, these power series representations are often used to define the exponential. I used the power series method to solve the differential equation y''+y=0 with y[0]=0 and y'[0]=1 using the following code. Print Book & E-Book. 5 lecture , §3. Linear methods applied to the solution of differential equations. The term "ordinary" is used in contrast with the term. Module 25 - Parametric. The Overflow Blog This week, #StackOverflowKnows molecule rings, infected laptops, and HMAC limits. Differential equation or system of equations, specified as a symbolic equation or a vector of | {
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Differential equation or system of equations, specified as a symbolic equation or a vector of symbolic equations. and then try to determine what the an. 4 Equations with Analytic Coefficients 453 *8. Back to top; 6: Power Series Solutions of Differential Equations; 6. Differential equation. In the first part of this course, the student learns to solve the most common types of differential equations. It is interesting to know whether sympy supports such equations along with usual ones. Even if this is the case, for simplicity we will see how the method works with a problem whose solution is a known function. Pourhabib Yekta1, A. Return to Differential Equations. 0012 Power Series Solution of Coupled Differential Equations in One Variable M. 1 in [EP], §5. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. equation, method of Laplace transforms for solving ordinary differential equations, series solutions (power series, Frobenius method); Legendre and Bessel functions and their orthogonal properties; Systems of linear first order ordinary differential equations. In the previous solution, the constant C1 appears because no condition was specified. Solve y0 = x2y with initial conditions y(0) = 1. The degree of an equation is the power to which the highest order term is raised. This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. which makes calculations very simple and interesting. 3 - Recursively Defined Sequences. We will then move to a problem whose solution can be expressed as a series only. Derivatives Derivative Applications Limits Integrals Integral Applications Series ODE Laplace Transform Taylor/Maclaurin Series Fourier Series. In the first part of this course, the student learns to solve the most common types of differential equations. S = dsolve(eqn) solves the differential equation eqn, where eqn is a symbolic equation. | {
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equations. S = dsolve(eqn) solves the differential equation eqn, where eqn is a symbolic equation. Various visual features are used to highlight focus areas. 1) The equation is linear of second order with polynomial coecients. EXAMPLE 2 Power Series Solution Use a power series to solve the differential equation Solution Assume that is. Notice that 0 is a singular point of this differential equation. A power series represents a function f on an interval of convergence, and you can successively differentiate the power series to obtain a series for and so. The general Airy differential equation is given by :$D^2y \pm m^2 x y = 0$or equivalently$y’’ \pm m^2 x y = 0$The differential equation in the question is a form of the Airy differential equation with the minus sign preceding $m^2$. Power Series Method for Nonlinear Partial Differential Equations Power series is an old technique for solving linear ordinary differential equations [7,20]. You can view the Laplace Table that you will be given on the Exam. 0012 Power Series Solution of Coupled Differential Equations in One Variable M. Power Series Solutions Differential Equations Power Series Solutions Differential Equations Yeah, reviewing a ebook Power Series Solutions Differential Equations could accumulate your near associates listings. Review of Series and Power Series. In this video we solve another differential equation by finding a power series solution. First order numerical / graphical differential equation solver: Transient analysis of RC or RL circuits. The solution diffusion. Each of the following waveform plots can be clicked on to open up the full size graph in a separate window. Solve y00 = xy0 +y with initial conditions y(0) = 1 and y0(0) = 0. AMS30, 151-156 (1971). Covers material on integration methods (trig, partial fractions, etc. The solution of the general differential equation dy/dx=ky (for some k) is C⋅eᵏˣ (for some C). This equation with concrete values of the parameter appeared in the articles | {
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is C⋅eᵏˣ (for some C). This equation with concrete values of the parameter appeared in the articles by F. DIFFERENTIAL EQUATIONS FOR ENGINEERS This book presents a systematic and comprehensive introduction to ordinary differential equations for engineering students and practitioners. We have step-by-step solutions for your textbooks written by Bartleby experts!. Laplace’s transformation maps a differential equation onto an algebraic equation, which can be solved relatively easily. Home » Supplemental Resources » Calculus Revisited: Complex Variables, Differential Equations, and Linear Algebra » Part II: Differential Equations » Lecture 6: Power Series Solutions Lecture 6: Power Series Solutions. SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS— SOME WORKED EXAMPLES First example Let’s start with a simple differential equation: ′′− ′+y y y =2 0 (1) We recognize this instantly as a second order homogeneous constant coefficient equation. Find materials for this course in the pages linked along the left. Series of Solutions Review of Power Series The Recurrence Power Solutions about an Ordinary Point Euler Equations Series Solutions Near a Regular Singular Point Equations of Hypergeometric Type Bessel’s Equations Legendre’s Equation Orthogonal Polynomials Review Questions for Chapter 6 Applications of Higher Order Differential Equations. Delay Differential Equations, Power Series, Taylor Series, Newton's Method 1. 7MB) To complete the reading assignments, see the Supplementary Notes in the Study Materials section. First‐order equations. In the next section, Section 8. Type: Artigo de periódico: Title: Group Classification Of A Generalized Black-scholes-merton Equation: Author: Bozhkov Y. This is a simple example and the final solution is very nice compared to what would normally happen with a more complicated differential equation, so please be aware of that!. order Power series solution of differential-algebraic equations in (1. My longest video yet, power series solution to | {
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solution of differential-algebraic equations in (1. My longest video yet, power series solution to differential equations, solve y''-2xy'+y=0, www. which makes calculations very simple and interesting. Power Series Method for Linear Partial Differential Equations of Fractional Order 73 Lemma 2. polynomial differential equations has a solution of this form. Differential equation. The ideas that you guess a power series solution to a differential equation and then you plug it in and in order to plug it in, you got to calculate its derivatives. First order numerical / graphical differential equation solver: Transient analysis of RC or RL circuits. The governing equation is also based on Kirchoff's law as described below. Back to top; 6: Power Series Solutions of Differential Equations; 6. Applications of first and second order equations; Sequences and infinite series. Write y' as dy/dx and the answer follows relatively - dy/dx + 3(a million+ x^2)y = 0 => dy/dx = -3(a million+x^2)y => dy/y = -3(a million+x^2)dx Now integrating the two components supplies ln(y) = -3x - x^3 + C (C is unknown integration consistent) => y = ok exp(-3x - x^3) the place ok = exp(C), and can be solved utilising extra suitable education approximately y (as an occasion the fee of y. {image} {image} {image} {image} 3. Making statements based on opinion; back them up with references or personal experience. Research Article Power Series Extender Method for the Solution of Nonlinear Differential Equations HectorVazquez-Leal 1 andArturoSarmiento-Reyes 2 Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. Yes, y(x) is the general solution of the differential equation represented as a power series. Example: an equation with the function y and its derivative dy dx. 3, Issue 4, April 2014 Solving the Systems of Differential Equations by a Power Series Method A. Differential Equations. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to | {
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Equations. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. Finding coefficients in a power series expansion of a rational function. 2 using Taylor series method of order four. Even if this is the case, for simplicity we will see how the method works with a problem whose solution is a known function. The TI-89 cannot solve second order linear differential equations with variable coefficients. This is the currently selected item. Solve the equation with the initial condition y(0) == 2. A differential equation (DE) is an equation relating a function to its derivatives. 1) The equation is linear of second order with polynomial coecients. Tìm kiếm power series solution of differential equations calculator , power series solution of differential equations calculator tại 123doc - Thư viện trực tuyến hàng đầu Việt Nam. Use MathJax to format equations. This note explains the following topics: First-Order Differential Equations, Second-Order Differential Equations, Higher-Order Differential Equations, Some Applications of Differential Equations, Laplace Transformations, Series Solutions to Differential Equations, Systems of First-Order Linear Differential Equations and Numerical Methods. This gives a recurrence formula for the coefficients. In the equation, represent differentiation by using diff. EXCHANGE RATE MISALIGNMENT AND CAPITAL INFLOWS: AN ENDOGENOUS THRESHOLD ANALYSIS FOR MALAYSIAABSTRACTThis study presents an attempt to investigate the impact of exchange rate misalignment on capital inflows in Malaysia. Power series solution (PSS) method is an old method that has been limited to solve linear differential equations, both ordinary differential equations (ODE) [1, 2] and partial differential equations (PDE) [3, 4]. m 0 Such a solution is obtained by substituting (2) and its derivatives into (1). = 1 for y at x = 1 with step length 0. For Example (i): $$\frac{d^3 x}{dx^3} + 3x\frac{dy}{dx} = e^y$$ In this equation | {
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with step length 0. For Example (i): $$\frac{d^3 x}{dx^3} + 3x\frac{dy}{dx} = e^y$$ In this equation the order of the highest derivative is 3 hence this is a third order. Next enter the c value and view the Laplace transform below the entry box. General Differential Equation Solver. Course summary; Differential equations Verifying solutions for differential equations: Series Power series intro: Series Function as a geometric series: Series Maclaurin series of eˣ, sin(x), and cos(x): Series Representing functions as power series: Series Telescoping series: Series Proof videos: Series. 2 - Series and Sequences of Partial Sums; Lesson 23. Initial conditions are also supported. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. 2 Power Series Section 5. compare the series solutions with the solutions of the differential equation obtained using the method of section 4. Calculator: A calculator such as TI83/84 plus will be allowed to use on tests and final exam. The first differential equation, , is rather easy to solve, we simply integrate both sides. Convergence of Sequences; Convergence of series using geometric series, the comparison tests, the alternating series test, the root test, and the ratio test. y'' − y' = 0. MA 401 Applied Differential Equations II (Wave, heat and Laplace equations. First, we present an introduction to Fourier Series, then we discuss how to solve differential equations using Fourier Series. In this work we present a power series method for solving ordinary and partial differential equations. Every project on GitHub comes with a version-controlled wiki to give your documentation the high level of care it deserves. It is licensed under the Creative Commons Attribution-ShareAlike 3. Exact Differential Equation Non-Exact Differential Equation M(x,y)dx+N(x,y)dy=0 N(x,y)y'+M(x,y)=0 Linear in x Differential Equation Linear in y Differential Equation RL Circuits Logistic Differential Equation | {
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x Differential Equation Linear in y Differential Equation RL Circuits Logistic Differential Equation Bernoulli Equation Euler Method Runge Kutta4 Midpoint method (order2) Runge Kutta23 2. 03/26/18 - We propose a computational method to determine when a solution modulo a certain power of the independent variable of a given algeb. The general Airy differential equation is given by :$D^2y \pm m^2 x y = 0$or equivalently$y’’ \pm m^2 x y = 0$The differential equation in the question is a form of the Airy differential equation with the minus sign preceding $m^2$. | {
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The common answers are: It cannot (and in some cases, shouldn't) be done. 0.4 Licence; 1 License; 2 Why RMarkdown. Expressions can also be used for titles, subtitles and x- and y-axis labels (but not for axis labels on persp plots). Some greek characters: $$\alpha \beta \Omega \omega$$ Some math characters: $$\times \div \int \sum \pm \infty \Re \angle \Longleftarrow$$ For Subscript: $$V_{AB} \times I_A$$ And Superscript: $$P = I^{2} \times R_{arm}$$ It had … You can simply use latex symbols with mathjax. be able to implement your own LaTeX settings in R Markdown and specifically in the R Package bookdown. This means that you can use all the standard Markdown syntax in addition to some LaTeX features that we will list below. share | improve this question | follow | edited Apr 16 '18 at 15:24. It is a combination of Markdown and LaTeX syntax, which creates a great writing experience for technical documents. For example, some people find it easier to use HTML tags for images. R Markdown for Scientists; About this. Any changes that occur in either your data set or the analysis are automatically updated in your document the next time … You can also use HTML code. symbols r knitr markdown syntax-highlighting. I would like to insert math heavy equation into r markdown file. Diese Liste mathematischer Symbole zeigt eine Auswahl der gebräuchlichsten Symbole, die in moderner mathematischer Notation innerhalb von Formeln verwendet werden. Use a productive notebook interface to weave together narrative text and code to produce elegantly formatted output. Use a productive notebook interface to weave together narrative text and code to produce elegantly formatted output. This is consistent now, but in contrast to mdmath versions prior to 2.0. Whenever I use some latex codes in my report and compile it in a HTML file, it does not get displayed correctly all the time. encoding is UTF-8, needs xelatex, like this:--- output: pdf_document: latex_engine: xelatex --- per mille sign. | {
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UTF-8, needs xelatex, like this:--- output: pdf_document: latex_engine: xelatex --- per mille sign. It’s a very simple language that allows you to write HTML in a shortened way. Cœur. Most of what is presented here isn’t primarily about how to use R, but rather how to work with tools in RMarkdown so that the final product is neat and tidy. add a comment | Your Answer Thanks for contributing an answer to Stack Overflow! R Markdown. Mathematical Annotation in R Description. Two topics that aren’t covered in the RStudio help files are how to insert mathematical text symbols and how to produce decent looking tables without too much fuss. Commonly used scientific symbols in pandoc markdown. January 5, 2019, 2:47am #1. Incorporating R results directly into your documents is an important step in reproducible research. Markdown which is a markup language that is a superset of HTML. Hi all, I always use R Markdown to create project reports, assignments, etc for my college homework. R Markdown supports a reproducible workflow for dozens of static and dynamic output formats including HTML, PDF, MS … This is not the same for PDF, which render correctly. LaTeX is especially useful when reports include scientific or mathematical symbols and notation. Yes. understand how R Markdown teams up with LaTeX to render math expressions. You can do this easily in R using the plotmath syntax. A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics. share | improve this question | follow | asked Mar 27 '19 at 16:50. iago iago. When you render, R Markdown 1. runs the R code, embeds results and text into .md file with knitr 2. then converts the .md file into the finished format with | {
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and text into .md file with knitr 2. then converts the .md file into the finished format with pandoc Set a document’s default output format in the YAML header:--- output: html_document --- # Body output value creates html_document html pdf_document pdf (requires Tex ) word_document Microso# Word (.docx) odt_document … For this guide, I’m going to use StackEdit . Latex to render mathematical and scientific writing. However the font will be in the mathjax font. Can I use math markup in code blocks ? Apparently, many others ( 1, 2, 3 ), have asked the same question. Using HTML is also helpful when you need to change the attributes of an element, like specifying the color of text or changing the width of an image. Markdown. plain text: ‰ (does render properly in PDF, does in Word) HTML: ‰ (does renders properly in PDF, does in Word) ex) $\sigma = \frac{1/\lambda}{\sqrt{n}}$ r math markdown. Microsoft Word does support large sized brackets, so I am unsure if this is expected behavior or a bug. It can be used on some websites like Stack Overflow or to write documentations (essentially on GitHub). share | improve this answer | follow | answered Mar 31 '19 at 18:18. itsmysterybox itsmysterybox. Turn your analyses into high quality documents, reports, presentations and dashboards with R Markdown. I have a bookdown book with a lot of greek symbols inline. Use multiple languages including R, Python, and SQL. 775 1 1 gold badge 8 8 silver badges 19 19 bronze badges. Yes. R Markdown adds a few features which include R code and results in the formatted document. This allow you write documents which integrate results from your analysis. Use the Markdown: Clip Markdown+Math to Html command or the key binding Ctrl+K.. If I render to gitbook or other HTML it looks as expected. 2.1 Overview; 2.2 Questions; 2.3 Objectives; 2.4 Your Turn; 2.5 Reproducibility is a problem; 2.6 Literate programming is a partial solution; 2.7 Markdown as a new player to legibility. The first official book | {
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is a partial solution; 2.7 Markdown as a new player to legibility. The first official book authored by the core R Markdown developers that provides a comprehensive and accurate reference to the R Markdown ecosystem. Can I use math markup in code blocks ? I need a tool that convert equation that convert into r markdown rules. Many Markdown applications allow you to use HTML tags in Markdown-formatted text. See all the Latex Math Symbols and difference between dot characters. Scroll down to #List of mathematical symbols for a complete list of Greek symbols. Provide details and share your research! List of Greek letters and math symbols. Ideally, what we want a free, cross-platform Markdown editor that comes packaged with the ability to input mathematical symbols, has support for tables, and allows for exporting as PDF. Da es praktisch unmöglich ist, alle jemals in der Mathematik verwendeten Symbole aufzuführen, werden in dieser Liste nur diejenigen Symbole angegeben, die häufig im Mathematikunterricht oder im Mathematikstudium auftreten. After reading this book, you will understand how R Markdown documents are transformed from plain text and how you may customize nearly every step of this processing. This book showcases short, practical examples of lesser-known tips and tricks to helps users get the most out of these tools. Use TinyTex. For contributing an answer to Stack Overflow or to write documentations ( essentially on ). Very well, but for a complete list of mathematical symbols and difference between dot characters your documents an! Markdown applications allow you write documents which integrate results from your analysis the require! ” and called the file rmdonrmd.Rmd specifically in the Mathjax font using the plotmath syntax into documents. Use the LaTeX math symbols and difference between dot characters your documents is an important step in reproducible research productive! The plotmath syntax to the R Markdown teams up with LaTeX to render math expressions | {
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# Domain of the n composed logarithms on x.
As we know, the real logarithm has the domain
$$D_1 = \{x : x \in \mathbb{R}, x > 0\}$$
What is the logarithmic domain of "higher order" logarithms, at index n? For example, it seems that
$$D_2 = \{x : x \in \mathbb{R}, x > 1\}$$
which would be the domain of
$$f(x) = \log(\log(x))$$
$D_3$ is a little hard to imagine. I think my real question deals with formalization, and the extended case of $D_n$ however.
There may be connections to the iterated logarithm, which says how many iterations are necessary before a value breaks in a certain base. See the wikipedia article.
• Yes, there is a connection. Which base did you have in mind? – hardmath Nov 14 '15 at 23:08
• base e? I was kind of asking abstractly, if you have an answer in an arbitrary base that would be impressive and helpful. – theREALyumdub Nov 14 '15 at 23:08
• Hint: If $\ln^{(n)}(x)$ denotes the iterated logarithm and ${^n}e=e^{e^{\ldots{e}}}$ ($n$-times) denotes tetration, then the following identity holds for all $n\in\mathbb{N}$: $\ln^{(n)}({^n}e)=1$. – Yiannis Galidakis Nov 14 '15 at 23:34
• @YiannisGalidakis, do you mean the super logarithm (inverse of tetration)? – theREALyumdub Nov 14 '15 at 23:37
• I think so, yes, although I am not very familiar with the actual definition Andrew Robbins uses. It's bullet 3 on that page :As the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm) @theREALyumdub – Yiannis Galidakis Nov 14 '15 at 23:42
If you denote by $\ln^{(n)}(x)$ the iterated logarithm and by ${^n}e=e^{e^{\ldots^{e}}}$ (height $n$) iterated exponentiation of the base $e$ (as per the comment), we have by definition:
$$\ln^{(n)}({^n}e)=1\Rightarrow$$ $$\ln^{(n+1)}({^n}e)=\ln(1)=0$$
Apply for $n=1$ and we get:
$$\ln(\ln(e))=0$$ | {
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Apply for $n=1$ and we get:
$$\ln(\ln(e))=0$$
So $D_2$ should be $D_2=\{x\in\mathbb{R}\colon x\ge e\}$. This however breaks the pattern, because the range of $\ln(\ln(x))$ can be negative for this case, if we extend the domain of $x$ to be $x\gt 1$.
You can't do this for higher iterates however, because negative ranges are not allowed in the domain of $\ln$. Therefore:
$$D_1=\{x\in\mathbb{R}\colon x\gt 0\}$$ $$D_2=\{x\in\mathbb{R}\colon x\gt 1\}$$ $$D_3=\{x\in\mathbb{R}\colon x\ge {^2}e\}$$ $$D_4=\{x\in\mathbb{R}\colon x\ge {^3}e\}$$
and in general ($n\ge 3$):
$$D_n=\{x\in\mathbb{R}\colon x\ge {^{n-1}}e\}$$
• Much more formal than my answer; I was so busy trying to edit mine to be correct that I missed this. – theREALyumdub Nov 15 '15 at 1:35
Actually, now that I've looked into it, the table on the page seems to suggest that the iterated logarithm values are the way you compute the bounds on the $D_n$ domains, and it switches at the tetration integer values.
## The tetration operation on the base defines the bounds of the domains $D_n$.
Let b be the base of the logarithm in question, so that $f_1(x) = \log_b(x)$
We can say
$$D_1 = \{x : x \in \mathbb{R}, x > 0\}$$ $$D_2 = \{x : x \in \mathbb{R}, x > b\}$$
And for n > 2, we apply the iterated logarithm to say that
$$D_3 = \{x : x \in \mathbb{R}, x > b^b\}$$ $$D_4 = \{x : x \in \mathbb{R}, x > (b^b)^b\}$$ $$...$$ $$D_n = \{x : x \in \mathbb{R}, x > tetra(b,n - 1)\}$$ | {
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• Yes, that looks ok. – Yiannis Galidakis Nov 15 '15 at 0:03
• The exponent should probably be $n-1$. Your $D_2$ is wrong, since $\ln(\ln(1))$ has problems. It should be $x>e$ for base $e$. Then $D_n=\{x:x\in\mathbb{R},x>tetra(e,n-1)\}$ for base $e$. – Yiannis Galidakis Nov 15 '15 at 0:14
• Does that look better? – theREALyumdub Nov 15 '15 at 1:32
• Your $D_2$ needs correction. It is $x>1$ (it is the only one that breaks the pattern, see my answer. My comment above has a typo) and when you notate tetration it's prettier to stick to one of the two forms you have: $tetra(b,3)=b^{b^{b}}$, but parentheses go from top to bottom. So it'd be $b^{(b^b)}$, $b^{(b^{(b^b)})}=tetra(b,4)$, etc. – Yiannis Galidakis Nov 15 '15 at 1:55 | {
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# A five digit number minus a four digit number equals $33333$. What are the two numbers, if you are only allowed to use the numbers $1-9$
Question: A five digit number minus a four digit number equals $$33333$$. What are the two numbers, if you are only allowed to use the numbers $$1-9$$ once? More precisely, \begin{align} & & A_1 \, A_2 \, A_3 \, A_4 \, A_5 \\ & - & A_6 \, A_7 \, A_8 \, A_9 \\ & & \hline 3\,\,\,\,\,3 \,\,\,\,\,3 \,\,\,\,\,\,3 \,\,\,\,\,\,3\\ & & \hline \end{align} where $$A_1,A_2,A_3,A_4,A_5, A_6, A_7, A_8, A_9 \in \{1,2,3,4,5,6,7,8,9\}$$ and they form a pairwise distinct set.
For me, I would guess $$A_1=3$$ or $$A_1 = 4.$$ But that is all I got. I am interested to know its thought process.
• Simplify the problem: A two digit number minus a one digit number equals $33$. What do you notice? Then move on to three digit minus two digit and so on. What are your thoughts on this? Mar 25 '20 at 4:51
• A quick Google search returns this reddit post from 6 years ago; it contains two answers but no solution. Mar 25 '20 at 4:53
• @AndrewChin For simpler problem like A two digit number minus a one digit number equals 33, what digits can I use? Mar 25 '20 at 5:16
• Note that these are digits, not numbers. Also, an individual $A_i$ cannot be pairwise distinct; it is the $A_i$ in the plural that are pairwise distinct. Mar 25 '20 at 6:50
• @joriki I edited my question, Is it okay now? Mar 25 '20 at 6:52
The sum of the digits $$1$$ through $$9$$ is odd. They contribute to the parity of the digit sum of the result no matter which row they’re in. The digit sum of the result is odd. Thus there must be an even number of borrowings.
A column that causes borrowing must have a $$7$$, $$8$$ or $$9$$ in the bottom row, so we cannot have four borrowings.
On the other hand, if there were no borrowing at all, the possible pairs in a column would be $$9-6-3$$, $$8-5-2$$ and $$7-4-1$$, but we can use at most one from each of these three groups. | {
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It follows that there are exactly two borrowings. Thus the difference between the digit sums of the rows must be $$5\cdot3-2\cdot9=-3$$, and since the sum of all digits is $$\frac{9(9+1)}2=45$$, the top row must sum to $$21$$ and the bottom row to $$24$$.
We need to have exactly two of $$7$$, $$8$$ and $$9$$ in the bottom row to cause the two borrowings.
It can’t be $$7$$ and $$8$$ because then $$7$$ would have to be subtracted from $$1$$ and $$8$$ from $$2$$, so the two borrowing columns would have to be the two lending columns.
If it were $$8$$ and $$9$$, that would leave a sum of $$7$$ for the bottom row, so that could be $$3,4$$ or $$2,5$$ or $$1,6$$. It can’t be $$3,4$$ because one of those needs to be $$A_1$$; it can’t be $$2,5$$ because $$5$$ would need to be subtracted from $$8$$ or $$9$$; and it can’t be $$1,6$$ because $$6$$ would need to be subtracted from $$9$$.
Thus $$7$$ and $$9$$ are in the bottom row. That leaves a sum of $$8$$ for the bottom row, which could be $$3,5$$ or $$2,6$$. But it can’t be $$2,6$$, again because $$6$$ would need to be subtracted from $$9$$.
Thus we have $$3,5,7,9$$ in the bottom row and $$1,2,4,6,8$$ in the top row. So $$4$$ must be $$A_1$$, $$7$$ must be subtracted from $$1$$, $$9$$ from $$2$$, $$3$$ from $$6$$ and $$5$$ from $$8$$. Thus the lenders must be $$4$$ and $$1$$, so the top row must start $$412$$. That leaves two possibilities for the order of the last two columns, so there are two solutions:
41286 41268
-7953 and -7935
----- -----
33333 33333
The solutions are confirmed by this Java code. (Full disclosure: I initially made a mistake in the proof and wrote the code to find it, so I knew the solution before I completed the proof.) | {
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• I do not fully understand the first paragraph. Why would the sum of digits from $1$ to $9$ contribute to the parity of the digit sum of the result? Mar 25 '20 at 12:15
• If you ignore borrowing for the time being, the digit sum of the result is a sum of the digits from $1$ to $9$, each with either a $+$ or a $-$. The sign makes no difference; in either case an odd digit contributes $1$ to the parity and an even digit contributes $0$. Mar 25 '20 at 12:19
• Can you eplain 'Thus the difference between the digit sums of the rows must be $5\cdot3-2\cdot9=-3$'. Where do we get $5\cdot3$ and $2\cdot9$? Mar 27 '20 at 7:02
• @Idonknow: There are five $3$s in the result, so the digit sum of the result is $5\cdot3$. If there were no borrowing, the digit sum of the result would be the difference of the digit sums of the rows. Each borrowing subtracts $1$ from the higher digit and adds $10$ to the lower digit, so the net effect is to add $9$ per borrowing. We found that there are exactly two borrowings. Thus the digit sum of the result is $2\cdot9$ more than the difference between the digit sums of the rows. Mar 27 '20 at 7:06 | {
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$\endgroup$ – André Nicolas Apr 30 '11 at 18:58 $\begingroup$ @shino: Or else if you are doing everything correctly, and exponential is a poor fit, look for a better fit from one of the Weibull distributions. The probability density function is $$f(x) = me^{-mx}$$. distribution is a discrete distribution closely related to the binomial distribution and so will be considered later. First consider λ = 1 λ = 1. Here e is the mathematical constant e that is approximately 2.718281828. Distributions with CV < 1 (such as an Erlang distribution ) are considered low-variance, while those with CV > 1 (such as a hyper-exponential distribution ) are considered high-variance [ … The ‘moment generating function’ of an exponential random variable X for any time interval t<λ, is defined by; M X (t) = λ/λ-t 0.5. c. 0.25. d. 2.0. e. the means of the two distributions can never be equal. Scientific calculators have the key “e … Therefore, X ~ Exp(0.25). Sample means from an exponential distribution do not have exponential distribution. The exponential distribution is one of the widely used continuous distributions. To describe the time between successive occurrences when all occurrences follow an exponential. μ = σ. Study notes and guides for Six Sigma certification tests. In fact, the mean and standard deviation are both equal to A. f ( x) = e-x/A /A, where x is nonnegative. Exponential Distribution Modelling Of Wet-day Rainfall Totals Assume An Exponential Distribution Can Be Used To Model Precipitation Totals On Wet Days. (Taken from ASQ sample Black Belt exam.). The distribution notation is X ~ Exp(m). Required fields are marked *. Exponential distribution is the time between events in a Poisson process. Exponential distribution is the time between events in a Poisson process. Construct a histogram of the dat The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by Simply, it is an inverse of Poisson. Log in or Sign up in | {
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X with rate parameter λ is given by Simply, it is an inverse of Poisson. Log in or Sign up in seconds with the buttons below! Exponential Distribution Variance. http://www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf, Your email address will not be published. The amount of time (may be in months) a car battery lasts. Learn how your comment data is processed. μ = σ. 100% of candidates who complete my study guide report passing their exam! Therefore, $$X \sim Exp(0.25)$$. This section requires you to be logged in. That is, for an Exponential distribution, the mean and the standard deviation are equal, and equal to the reciprocal of the rate parameter. Questions, comments, issues, concerns? E ( X k) = ∫ 0 ∞ x k e − x d x = k! The mean of exponential distribution is 1/lambda and the standard deviation is also 1/lambda. This section requires you to be a Pass Your Six Sigma Exam member. (. it describes the inter-arrival times in a Poisson process.It is the continuous counterpart to the geometric distribution, and it too is memoryless.. This statistics video tutorial explains how to solve continuous probability exponential distribution problems. It can be shown for the exponential distribution that the mean is equal to the standard deviation; i.e., μ = σ = 1/λ Moreover, the exponential distribution is the only continuous distribution that is I have seen this question on one of the websites (I guess ASQ, not sure). 16. Therefore, X ~ Exp(0.25). IASSC Lean Six Sigma Green Belt Study Guide, Villanova Six Sigma Green Belt Study Guide, IASSC Lean Six Sigma Black Belt Study Guide, Villanova Six Sigma Black Belt Study Guide, Where e is base natural logarithm = 2.71828. I’ll investigate the … If it is a negative value, the function is zero only. $$\mu = \sigma$$ The distribution notation is $$X \sim Exp(m)$$. Your instructor will record the amounts in dollars and cents. If the number of occurrences follows a Poisson distribution, the lapse of time between these events is | {
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the number of occurrences follows a Poisson distribution, the lapse of time between these events is distributed exponentially. Login to your account OR Enroll in Pass Your Six Sigma Exam. It is a number that is used often in mathematics. The exponential distribution can be used to determine the probability that it will take a given number of trials to arrive at the first success in a Poisson distribution; i.e. This site uses Akismet to reduce spam. | {
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.
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# How can I write this power series as a power series representation?
How can I write this power series ($1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8....$) as a power series representation (like $\dfrac{1}{1-x}$ or something neat like that)?
• This is S(1+x) where $S=1+2x^2+3x^4+...$ for a start. – Paul Dec 19 '14 at 23:18
• @paul: What about x? – Mathy Person Dec 19 '14 at 23:21
• @Paul Saying $(1+x)S(x)$ might be more clear. – Tim Raczkowski Dec 19 '14 at 23:25
Hint: using $y=x^2$ and derivative in $y$: $$(1+x)(1+2x^2+3x^4+\ldots)$$ $$=(1+x)(1+2y+3y^2+4y^3 +\ldots)$$ $$= (1+x)(y+y^2+y^3+y^4+\ldots)'$$ $$= (1+x)\left( \frac{y}{1-y}\right)'$$
Edit: $$= (1+x) \frac{1}{(1-y)^2}$$ $$= \frac{1+x}{(1-x^2)^2}$$ $$= \frac{1}{(1-x)(1+x^2)}.$$
• I believe $(y+y^2+y^3+y^4+....)$ = $\frac{1}{1-y}$, if I am not mistaken. Then is it: $\frac{1+x}{1-y}$? – Mathy Person Dec 19 '14 at 23:28
• Oh wait, I see that you had $\frac{y}{1-y}$ instead. Is it: $\frac{(1+x)(y)}{1-y}$? – Mathy Person Dec 19 '14 at 23:30
• And $\frac{(1+x)(y)}{1-y}$, and $y=x^2$, then $\frac{(1+x)(x^2)}{1-x^2}$? – Mathy Person Dec 19 '14 at 23:30
• Simplifying would result in: $\frac{x^2}{1-x}$? – Mathy Person Dec 19 '14 at 23:31
• Just take derivative in y of $y/(1-y)$, then replace $y$ with $x^2$. – ir7 Dec 19 '14 at 23:33
I would go about this by first splitting the series up:
$$1+x+2x^2+2x^3+3x^4+3x^5+...=(1+x)(1+2x^2+3x^4+...)$$
Letting $s=1+2x^2+3x^4$ we can do a few tricks:
$$s-x^2s=\begin{array}{c} 1&+2x^2&+3x^4+... \\ &-x^2&-2x^4-...\end{array}$$ $$=1+x^2+x^4+...$$
Which converges to $\frac{1}{1-x^2}$ for $-1 < x < 1$ (proving this is not hard, and can be done by a technique like the above). This gives
$$s -x^2s=\frac{1}{1-x^2}\Leftrightarrow s=\frac{1}{(1-x^2)^2}=$$
Thus the original series converges to:
$$(1+x)s=\frac{(1+x)}{(1-x^2)^2}$$
For $-1 < x - 1$. | {
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Thus the original series converges to:
$$(1+x)s=\frac{(1+x)}{(1-x^2)^2}$$
For $-1 < x - 1$.
• Hmm..I got $\frac{x^2}{1-x}$ (see what I posted in reply to ir7's comment). Did I make a mistake somewhere? – Mathy Person Dec 19 '14 at 23:33
• Whoops my bad, I forgot a step – SBareS Dec 19 '14 at 23:36
• Apparently I am to tired for maths right now... – SBareS Dec 20 '14 at 0:05
Unless I'm mistaken, it is $$\sum_{n=1}^\infty nx^{2n-2} + \sum_{n=1}^\infty nx^{2n-1}$$ If you can compute one of the two terms, e.g. $\sum_{n=1}^\infty nx^{2n-2} = \sum_{n=0}^\infty (n+1)x^{2n} = \sum_{n=0}^\infty (n+1)(x^{2})^n$ (see e.g. this, then you'll also get the other term (by multiplying it by $x$), and thus the sum.
One way is to look at $$1+2x^2+3x^4+4x^6+5x^8+\dots$$ as $$1+2t+3t^2+4t^3+5t^4+\dots$$ where $t=x^2$. The last series is the derivative of $$1+t+t^2+t^3+t^4+t^5+\dots=\frac1{1-t}$$ Therefore, $$1+2t+3t^2+4t^3+5t^4+\dots=\frac1{(1-t)^2}$$ and $$1+2x^2+3x^4+4x^6+5x^8+\dots=\frac1{(1-x^2)^2}$$ Now, just multiply by $1+x$: \begin{align} 1+x+2x^2+2x^3+3x^4+3x^5+4x^6+4x^7+5x^8+5x^9+\dots &=\frac{1+x}{(1-x^2)^2}\\ &=\frac1{(1-x)(1-x^2)} \end{align} | {
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# How to expand binomials?
I'm working on a few proofs and am missing how this algebra works....
So, how does one expand $(k+1)^3\,$? Can I use FOIL? What does it expand to?
And how to expand $(k+1)^5\,$?
Thanks!
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look up Pascal's Triangle, and Binomial Theorem. Also no, you can't use FOIL, at very least you must use distributive property. – Joseph Skelton Jan 22 '13 at 20:59
Check out the entry Binomial Theorem in Wikipedia.
Putting $y = 1$, that will give you the tools you need to expand $(k+1)^3,\; (k+1)^5, \;$ and $\,(k + 1)^n\,$ for any non-negative integer $n$.
FOIL works fine for $(k + 1)^2 = k^2 + 2k + 1$
One can go a step further by distributing $(k+1)$ over $(k^2 + 2k + 1)$ to get $$(k^3 + 3k^2 + 3k + 1) = (k + 1)^3.$$
But for large exponents, it's handy to know the pattern of coefficients that correspond to different powers of $k$ in the expansion of $(k+1)^n$: Pascal's triangle shows this handy relationship.
I'll include an animation and image of "Pascal's Triangle" which displays the coefficients of expansions of a binomial $(k + 1)$ (these coefficients are referred to as: binomial coefficients): up to and including fourth and fifth degree binomials, respectively:
$\quad\quad\quad\quad$ $\quad\quad\quad\quad$
$$\text{Each number in the triangle is the sum of the two directly above it.}$$
To see how this "plays out" in the expansion of $(x + 1)^n,\;0 \le n \le 6$: | {
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To see how this "plays out" in the expansion of $(x + 1)^n,\;0 \le n \le 6$:
$$(x + 1)^0 = \color{blue}{\bf{1}}$$ $$(x + 1)^1 = \color{blue}{\bf{1}}\cdot x +\color{blue}{\bf{1}}$$ $$(x + 1)^2 = \color{blue}{\bf{1}}\cdot x^2 + \color{blue}{\bf{2}}x + \color{blue}{\bf{1}}$$ $$(x+1)^3 = \color{blue}{\bf{1}}\cdot x^3 + \color{blue}{\bf{3}}x^2 + \color{blue}{\bf{3}}x + \color{blue}{\bf{1}}$$ $$(x+1)^4 = \color{blue}{\bf{1}}\cdot x^4 + \color{blue}{\bf{4}} x^3+ \color{blue}{\bf{6}}x^2 + \color{blue}{\bf{4}}x +\color{blue}{\bf{1}}$$ $$(x+1)^5 = \color{blue}{\bf{1}}\cdot x^5 + \color{blue}{\bf{5}}x^4 + \color{blue}{\bf{10}} x^3 + \color{blue}{\bf{10}} x^2 + \color{blue}{\bf{5}}x + \color{blue}{\bf{1}}$$ $$(x + 1)^6 = \color{blue}{\bf{1}}\cdot x^6 + \color{blue}{\bf{6}}x^5 +\color{blue}{\bf{15}}x^4 + \color{blue}{\bf{20}}x^3 +\color{blue}{\bf{15}}x^2 + \color{blue}{\bf{6}}x + \color{blue}{\bf{1}}$$ $${\bf{\vdots}}$$
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Image source: Wikipedia, found at link to "Pascal's Triangle". – amWhy Jan 22 '13 at 22:17
Nice animated GIF! The answer is good, too :-) (+1) – robjohn Jan 22 '13 at 22:53
Thanks, @robjohn! It says more concisely, in animation, what would take a bit of explaining to write! – amWhy Jan 22 '13 at 22:56
@amWhy: It is unbelievable!! Great!! I cannot say anything. Nothing such this can describe the problem soooo well. +10! – Babak S. Jan 23 '13 at 15:50
This is incredible! Perfect explanation. Thank you! – user56763 Jan 23 '13 at 16:44 | {
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Before you jump to the binomial theorem (still the best way to go, in general, for expressions of the form $(a+b)^n$), let's start at the beginning. You undoubtedly know that $$(k+1)^3=(k+1)(k+1)(k+1)$$ We'll start by expanding $(k+1)(k+1)$. You can use FOIL here so we have $$(k+1)(k+1)=k^2+2k+1$$ We're two-thirds of the way to the answer. Now we have $$(k+1)^3=(k+1)(k^2+2k+1)$$ and by the distributive property, namely that $(a+b)c=ac+bc$, we have \begin{align} (k+1)^3=(k+1)(k^2+2k+1)&=(k)(k^2+2k+1)+(1)(k^2+2k+1)\\ &=(k^3+2k^2+k)+(k^2+2k+1)\\ &=k^3+3k^2+3k+1 \end{align} This will work for any positive integer exponent but, as Michael notes, you wouldn't want to do this for $(a+b)^n$.
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Thanks, makes sense. – user56763 Jan 23 '13 at 16:45
@ user56763: From what Rick Decker posted you can see that $(k+1)^3 = (k+1)(k+1)(k+1)$ can be expanded in the following way. Write down all possible 3-fold products of the form $ABC,$ where $A$ is a choice of one of the terms in the first $(k+1)$ factor, $B$ is a choice of one of the terms in the second $(k+1)$ factor, and $C$ is a choice of one of the terms in the third $(k+1)$ factor. Then add the 8 products. Thus, among the things you'll add will be $(k)(k)(1)$ and $(1)(k)(k)$ and $(k)(1)(k).$ Note that there are 3 of these with exactly two $k$'s and exactly one $1,$ so you get $3k^2$ ... – Dave L. Renfro Jan 28 '13 at 19:30
You can use "FOIL" twice. You should get $$k^3+3k^2+3k+1.$$ More generally, $$(a+b)^3 = a^3 + 3a^2b+3ab^2+ b^3.$$
Please don't vacilate between lower-case $k$ and capital $K$ in mathematical notation. Pick one and stick to it. Mathematical notation is case sensitive. Sometimes one uses lower-case $k$ and capital $K$ for two different things in the same problem, and you need to be clear about which is which. | {
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But it would take a while to use FOIL to get $$(a+b)^9 = a^9+9a^8b+36a^7b^2+84a^6b^3+126a^5b^4+126a^4b^5+84a^3b^6+36a^2b^7+9ab^8+b^9.$$ That's one reason to be aware of the binomial theorem, which explains the pattern.
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What you're looking for is the binomial theorem, where y = 1.
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This is very scant. Something more would be good: some explanation or at least a link to an article. Of course, an answer should be more than simply a link. Look at the other answers for examples. – robjohn Jan 22 '13 at 22:56
Yeah, I was going to, but was having trouble typing out the math for a more exhaustive answer. – MITjanitor Jan 23 '13 at 2:10 | {
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# This is not a necessary part of the course i'm taking, but an interesting problem.
## Homework Statement
Let a; b; c $\in$ (1,∞) and m; n $\in$ (0,∞). Prove that
\log_{b^mc^n} a + \log_{c^ma^n} b +\log_{a^mb^n} c \ge \frac 3 {m + n}
## The Attempt at a Solution
I do not even know where to start. A coherent explanation and possible solutions would greatly farther my knowledge of mathematics. Thanks for any and all help.
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Do you mean $$\log_{b^mc^n} a + \log_{c^ma^n} b +\log_{a^mb^n} c \ge \frac 3 {m + n}$$
Or something else?
And what does "a; b; c (1;1) and m; n (0;1)" mean?
To your first post yes and thank you. Also, Let a; b; c $\in$ (1,∞) and m; n $\in$ (0,∞).
Last edited:
SammyS
Staff Emeritus
Homework Helper
Gold Member
## Homework Statement
Let a; b; c $\in$ (1,∞) and m; n $\in$ (0,∞). Prove that
$\log_{b^mc^n} a + \log_{c^ma^n} b +\log_{a^mb^n} c \ge \frac 3 {m + n}$
## The Attempt at a Solution
I do not even know where to start. A coherent explanation and possible solutions would greatly farther my knowledge of mathematics. Thanks for any and all help.
See how you might use the "Change of base" formula:
$\displaystyle \log_T(P)=\frac{\log_R(P)}{\log_R(T)}=\frac{\log_{10}(P)}{\log_{10}(T)}=\frac{\ln(P)}{\ln(T)}$
I think you could use this property $$\log_b x = \frac {\log_k x} {\log_k b}$$ Using that, for example, $$\log_{b^mc^n} a = \frac {\log_a a}{\log_a b^mc^n} = \frac {1}{\log_a b^mc^n}$$ You could deal with the other logs similarly, so you would get some rational expression involving sums and products of $m\log_a b$ with permutations of a, b, c, m and n. Another consequence of the formula above is that $$\log_b a = \frac {\log_a a} {\log_a b} = \frac 1 {\log_a b}$$ so you should be able to express everything in terms of $A = \log_a b$, $B = \log_a c$ and $C = \log_b c$ and m and n. After some algebra (probably, quite some algebra) you should be able to simplify that into something manageable and prove the inequality. | {
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Well, that certainly clears a lot up for me, but I would very much appreciate some examples of where to go from there. (As you can tell, I'm clearly not versed well enough in mathematics to attempt such a problem). Thank you for any more input and helping me get a better grasp on mathematics.
SammyS
Staff Emeritus
Homework Helper
Gold Member
M4th,
Where did you get this problem from? What is the level of the mathematics that might be used to solve this problem?
BTW: Notice what you get if a = b = c .
I am in a pre-calculus course but my teacher chose me to be part of a separate "group" of students who solves problems outside of the classroom. I would just like to be able to simply understand what is going on in some of these problems.
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Using the properties of the logarithm as we already discussed, you can show that $$\\ \log_{b^mc^n} a + \log_{c^ma^n} b + \log_{a^mb^n} c = \frac {1} {mA + nB} + \frac {1} {mB/A + n/B} + \frac {1} {m/A + nA/B}$$ where $$1 \le A \le B$$ Then you will need to prove $$\frac {1} {mA + nB} + \frac {1} {mB/A + n/B} + \frac {1} {m/A + nA/B} \ge \frac 3 {m + n}$$
This is a great forum, I already have learned a bit more about logarithms just watching how you've broken this problem up into its subsequent pieces. I thank you for such rapid responses and hospitality. Im looking forward to learning more about mathematics from you all and would greatly appreciate some more information on this and many more problems to come in my quest to broaden my grasp on this subject. I would also greatly appreciate a worded summary as to how you get to these results. I understand alot of it, but am still missing some peices and any more details would be way more than I expected to get but would also be extremely helpful.
Thank you again | {
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Thank you again
This forum is great because it helps you understand the problem and learn how to solve it. Learn, but not just copy a complete solution. That would be against the rules. The rules require that you make an attempt at solving the problem. Here that would be trying to get the expression to the form given above. You have been given enough information to do so. Now you should try and tell us whether that worked and when not, what exactly went wrong.
Yes my point exactly. My goal is to be able to arrive at, and transform equations with ease into more workable ones as you have above and really have a good grasp at how to do so with any other similar problems.
Now one problem in my understanding is I grasp how logba=logaa/logab and thus I see where
\\log_{b^mc^n} a = \frac {\log_a a}{\log_a b^mc^n} = \frac {1}{\log_a b^mc^n}
but how is that transformed into the last mathematical statement you made
explain how you can transform logb^mc^na = 1/logabmcn into the statement below
Using the properties of the logarithm as we already discussed, you can show that $$\\ \log_{b^mc^n} a + \log_{c^ma^n} b + \log_{a^mb^n} c = \frac {1} {mA + nB} + \frac {1} {mB/A + n/B} + \frac {1} {m/A + nA/B}$$ where $$1 \le A \le B$$ Then you will need to prove $$\frac {1} {mA + nB} + \frac {1} {mB/A + n/B} + \frac {1} {m/A + nA/B} \ge \frac 3 {m + n}$$
explain how you can transform logb^mc^na = 1/logabmcn into the statement below
Basic logarithm properties:
$$\log xy = \log x + \log y \\ \log x^y = y \log x$$
Apply them to $\log_a b^mc^n$ and see where that gets you.
Of course, now it's all starting to come together. Thanks again for your help. | {
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Cardinality and ordinality Purplemath. We attempt to follow the standards set out in Norman Weinberg’s Guide to Standardized Drumset Notation. If you want to get in on their secrets, you'll want to become familiar with these Venn diagram symbols. Note that it's unnecessary to load amsmath if you load mathtools. The following table gives a summary of the symbols use in sets. While crow's foot notation is often recognized as the most intuitive style, some use OMT, IDEF, Bachman, or UML notation, according to their preferences. Set Theory • A mathematical model that we will use often is that of . Compact set notation is a useful tool to describe the properties of each element of a set, rather than writing out all elements of a set. Sometimes the set is written with a bar instead of a colon: {x¦ x > 5}. Basic set notation. State whether each … Because null set is not equal to A. Basic set operations. Thankfully, there is a faster way. Inequalities can be shown using set notation: {x: inequality}where x: indicates the variable being described and inequality is written as an inequality, normally in its simplest form. Typing math symbols into Word can be tedious. Basic Set Theory . The Universal Set … Set Notation. Which is why the bulk of this follow-up piece covers the very basics of set theory notation, operations & visual representations extensively. Also, check the set symbols here.. ... Set Language And Notation. This section is to introduce the notation to the reader and explain its usage. Illustration: In this section, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three ways, inequality notation, set-builder notation, and interval notation. Use set notation to describe: (a) the area shaded in green (b) the area shaded in red : Look at the venn diagrams on the left. Symbol Symbol Name Meaning / definition Example { } set: a collection of elements: A = {3,7,9,14}, Set notation and Venn diagrams | {
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definition Example { } set: a collection of elements: A = {3,7,9,14}, Set notation and Venn diagrams questions. On the Insert tab, in the Symbols group, click the arrow under Equation, and then click Insert New Equation. Lots symbols look similar but mean different things. The guide you are now reading is a “legend” to how we notate drum and percussion parts when we engrave music at Audio Graffiti. Symbolab: equation search and math solver - solves algebra, trigonometry and calculus problems step by step Example: Set-Builder Notation: Read as: Meaning: 1 {x : x > 0}the set of all x such that x is greater than 0. any value greater than 0: 2 {x : x ≠ 11}the set of all x such that x is any number except 11. any value except 11: 3 {x : x < 5}the set of all x such that x is any number less than 5. any value less than 5 Use set notation to describe: (a) the area shaded in blue (b) the area shaded in purple. Email. The domains and ranges used in the discrete function examples were simplified versions of set notation. Sets, in mathematics, are an organized collection of objects and can be represented in set-builder form or roster form.Usually, sets are represented in curly braces {}, for example, A = {1,2,3,4} is a set. Universal set and absolute complement. Hyperbolic functions The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Author: Created by Maths4Everyone. Set notation practice. The table below lists all of the necessary symbols for compact set notation. The default way of doing it is to use the Insert > Symbols > More Symbols dialog, where you can hunt for the symbol you want. Mathematical Set Notation. Shading task. elements . When using set notation, we use inequality symbols to describe the domain and range as a set of values. This quiz and attached worksheet | {
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symbols to describe the domain and range as a set of values. This quiz and attached worksheet will help gauge your understanding of set notation. Below is the complete list of Windows ALT codes for Math Symbols: Set Membership & Empty Sets, their corresponding HTML entity numeric character references, and when available, their corresponding HTML entity named character references, and Unicode code points. There are many different symbols used in set notation, but only the most basic of structures will be provided here. They are { } and { 1 }. Crow's foot notation, however, has an intuitive graphic format, making it the preferred ERD notation for Lucidchart. Look at the venn diagram on the left. Click the arrow next to the name of the symbol set, and then select the symbol set that you want to display. Let us discuss the next stuff on "Symbols used in set theory" If null set is a super set Topics you will need to know in order to pass the quiz include sets, subsets, and elements. Let’s kick off by introducing the two most basic symbols for notating a set & it’s corresponding elements. Null set is a proper subset for any set which contains at least one element. Set notation is an important convention in computer science. Here null set is proper subset of A. For example, a set F can be specified as follows: = {∣ ≤ ≤}. take the previous set S ∩ V ; then subtract T: This is the Intersection of Sets S and V minus Set T (S ∩ V) − T = {} Hey, there is nothing there! In set-builder notation, the set is specified as a selection from a larger set, determined by a condition involving the elements. Subset, strict subset, and superset. Set notation. MS Word Tricks: Typing Math Symbols 2015-05-14 Category: MS Office. mathematical sets • A (finite) set can be thought of as a collection of zero or more . of . The Wolfram Language has the world's largest collection of consistent multifont mathematical notation characters\[LongDash]all fully integrated into both typesetting and | {
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multifont mathematical notation characters\[LongDash]all fully integrated into both typesetting and symbolic expression construction . A set is a well-defined collection of distinct objects. Consider the set $\left\{x|10\le x<30\right\}$, which describes the behavior of $x$ in set-builder notation. Preview. Set notation is used to help define the elements of a set. Set theory is one of the foundational systems for mathematics, and it helped to develop our modern understanding of infinity and real numbers. Under Equation Tools , on the Design tab, in the Symbols group, click the More arrow. It is still a set, so we use the curly brackets with nothing inside: {} The Empty Set has no elements: {} Universal Set. Intersection and union of sets. You never know when set notation is going to pop up. Sets. For example, let us consider the set A = { 1 } It has two subsets. Bringing the set operations together. Usually, you'll see it when you learn about solving inequalities, because for some reason saying "x < 3" isn't good enough, so instead they'll want you to phrase the answer as "the solution set is { x | x is a real number and x < 3 }".How this adds anything to the student's understanding, I don't know. Set theory starter. Set Theory Symbols Posted in engineering by Christopher R. Wirz on Wed Feb 08 2017. This carefully selected compilation of exam questions has fully-worked solutions designed for students to go through at home, saving valuable time in class. The following list documents some of the most notable symbols in set theory, along each symbol’s usage and meaning. The individual objects in a set are called the members or elements of the set. They wrote about it on the chalkboard using set notation: P = {Kyesha, Angie and Eduardo} When Angie's mother came to pick her up, she looked at the chalkboard and asked: What does that mean? If you … ALT Codes for Math Symbols: Set Membership & Empty Sets Read More » Admin Igcse Mathematics Revision Notes, O Level | {
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Symbols: Set Membership & Empty Sets Read More » Admin Igcse Mathematics Revision Notes, O Level Mathematics Revision Notes 2 Comments 12,074 Views. In, sets theory, you will learn about sets and it’s properties. When picking a symbol, best to trust the symbol's unicode name for its meaning, not appearance. Occasionally we will introduce a new symbol to cater for an unusual requirement of a client. Created: Jan 19, 2018 | Updated: Feb 6, 2020. Symbols Used in this Book; Glossary; While a comprehensive list of notation is included in the appendix, that is meant mostly as a reference tool to refresh the reader of what notation means. Researchers and mathematicians have developed a language and system of notation around set theory. To fully embrace the world of professional Venn diagrams, you should have a basic understanding of the branch of mathematical logic called ‘set theory’ and its associated symbols and notation. The table below contains one example set… 8 February 2019 OSU CSE 1. Solution: Let P be the set of all members in the math Problem 1: Mrs. Glosser asked Kyesha, Angie and Eduardo to join the new math club. A variant solution, also based on mathtools, with the cooperation of xparse allows for a syntax that's closer to mathematical writing: you just have to type something like\set{x\in E;P(x)} for the set-builder notation, or \set{x_i} for sets defined as lists. Google Classroom Facebook Twitter. The colon means such that.. For example: {x: x > 5}.This is read as x such that x is greater than > 5.. and symbols. Some notations for sets are: {1, 2, 3} = set of integers greater than 0 … Because rarely used symbol may look very different on another computer. Probability and statistics symbols table and definitions - expectation, variance, standard deviation, distribution, probability function, conditional probability, covariance, correlation After school they signed up and became members. CCSS.Math: HSS.CP.A.1. Quiz & Worksheet Goals A set is a collection of | {
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signed up and became members. CCSS.Math: HSS.CP.A.1. Quiz & Worksheet Goals A set is a collection of objects, things or symbols which are clearly identified.The individual objects in the set are called the elements or members of the set. Demo. In this notation, the vertical bar ("|") means "such that", and the description can be interpreted as "F is the set of all numbers n, such that n is an integer in the range from 0 to 19 inclusive". IGCSE 9-1 Exam Question Practice (Sets + Set Notation) 4.9 34 customer reviews. Relative complement or difference between sets. The symbols shown in this lesson are very appropriate in the realm of mathematics and in mathematical logic. That is OK, it is just the "Empty Set". S et theory is a branch of mathematics dedicated to the study of collections of objects, its properties, and the relationship between them. This cheat sheet is extremely useful. any. Mathematical logic the symbols use in sets to go through at home, saving valuable time class! Unusual requirement of a colon: { x¦ x > 5 } of notation around set,... May look very different on another computer Kyesha, Angie and Eduardo to join new...: ( a ) the area shaded in blue ( b ) the shaded! Of exam questions has fully-worked solutions designed for students to go through at home, saving valuable time class! Using set notation is going to pop up Notes, O Level Revision. With these Venn diagram symbols attempt to follow the standards set out in Norman Weinberg s. • a mathematical model that we will use often is that of domain and range as a of! 2018 | Updated: Feb 6, 2020 diagram symbols used symbol may look different... Empty set '' when set notation is going to pop up is OK, it is just the Empty! Empty set '' it helped to develop our modern understanding of and. To pop up out in Norman Weinberg ’ s properties condition involving the.... The arrow next to the reader and explain its usage notable symbols in set theory one... Trust the symbol 's unicode name for its meaning, | {
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will be here! Infinity and real numbers never know when set notation is going to pop up unusual requirement a... To pass the quiz include sets, subsets, and elements with these Venn symbols! One of the necessary symbols for compact set notation is going to pop up that it 's to. Symbols group, click the arrow next to the reader and explain its usage: Typing Math 2015-05-14. Us consider the set is written with a bar instead of a colon: { x... Through at home, saving valuable time in class shaded in purple symbols group, click the more arrow may!, but only the most basic of structures will be provided here more arrow get in on their secrets you. Following table gives a summary of the most basic of structures will be here. Going to pop up will introduce a new symbol to cater for an unusual requirement a. Admin Igcse Mathematics Revision Notes, O Level Mathematics Revision Notes 2 Comments 12,074.. Carefully selected compilation of exam questions has fully-worked solutions designed for students to go at... The necessary symbols for compact set notation, operations & visual representations extensively has an intuitive format... Set a = { ∣ ≤ ≤ } domain and range as a collection of distinct objects researchers and have... Important convention in computer science, you 'll want to get in on their secrets, you 'll to. Mathematics and in mathematical logic only the most basic of structures will be provided here, appearance! Helped to develop our modern understanding of infinity and real numbers symbol to cater for an unusual requirement of colon! S usage and meaning off by introducing the two most basic symbols for compact set notation Venn. Visual representations extensively preferred ERD notation for Lucidchart set '' is why the bulk of follow-up. Problem 1: Mrs. Glosser asked Kyesha, Angie and Eduardo to join the new Math.. If you want to become familiar with these Venn diagram symbols ∣ ≤. Off by introducing the two most basic of structures will be provided here the two | {
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symbols ∣ ≤. Off by introducing the two most basic of structures will be provided here the two most of... Bar instead of a client used in set notation is going to pop up and real numbers range a... Mathematics Revision Notes, O Level Mathematics Revision Notes 2 Comments 12,074 Views just the Empty... Of Mathematics and in mathematical logic list documents some of the set a = { 1 } it two! Set is written with a bar instead of a client at home, saving valuable time in.... X¦ x > 5 } Tricks: Typing Math symbols 2015-05-14 Category: ms Office will provided... It helped to develop our modern understanding of infinity and real numbers thought of as a selection from a set. Using set notation symbols notation, operations & visual representations extensively sets and it helped to develop modern... To the reader and explain its usage { ∣ ≤ ≤ } selected compilation exam! Set of values Equation Tools, on the Design tab, in the shown. The area shaded in purple get in on their secrets, you will about! Inequality symbols to describe: ( a ) the area shaded in blue ( b the... To develop our modern understanding of infinity and real numbers in on secrets... Off by introducing the two most basic symbols for notating a set written! Ms Word Tricks: Typing Math symbols 2015-05-14 Category: ms Office in blue ( b ) the shaded! Preferred ERD notation for Lucidchart in on their secrets, you will need know! To become familiar with these Venn diagram symbols developed a language and system of notation set! Will be provided here the foundational systems for Mathematics, and then select the symbol set that you to... ≤ ≤ } appropriate in the discrete function examples were simplified versions of set theory its meaning not! Using set notation is an important convention in computer science Igcse Mathematics Revision Notes, O Mathematics... Load mathtools ms Word Tricks: Typing Math symbols 2015-05-14 Category: Office., operations & visual representations extensively Empty set '' graphic format, | {
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Category: Office., operations & visual representations extensively Empty set '' graphic format, making it the preferred ERD for. 19, 2018 | Updated: Feb 6, 2020 be thought of as a selection from a set... S kick off by introducing the two most basic of structures will be provided here Weinberg ’ s.... Join the new Math club a larger set, and elements symbols use sets. | {
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# Why do these two methods of calculating the probability of winning a best-of-7 series give the same answer?
I was having a discussion with a friend about the probability, and we came up with very different methods to solve it that lead to the same answer. The problem is pretty simple: you have two teams A and B playing a best of 7 series where wins are independent and the probability that team B wins is .35. What's the probability of team B winning th series? Friend says, "this is just a binomal random variable $X$ with $n=7$ and $p=.35$ and we are looking for $P(X\geq 4)$, which according to my TI-83 here is .1998".
I was rather convinced that this must be wrong. If we think of the series as a sequence of A's and B's, then we are looking for the probability of obtaining a sequence of length $n=4,5,6,7$ with 4 B's and the last element is a B. I was sure that his method will include the probability of obtaining, say, {B,A,B,B,B,A,A}, which we are not interested in and in fact couldn't even ever occur. So I figure that what we really want for a sequence of length $n$ is the probability of obtaining a sequence of length $n-1$ with exactly 3 B's, and then tacking a B onto the end. So for a sequence of length $n$, the probability should be $\binom{n-1}{3}.35^4 .65 ^{n-4}$, and then the answer should be $\sum_{n=4}^{7}\binom{n-1}{3}.35^4 .65 ^{n-4}$. I was confident that I was right and he was wrong, but then I plugged that into Wolfram Alpha and got... .1998.
What's going on here? Is it a coincidence? | {
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What's going on here? Is it a coincidence?
-
Why do you say that {B,A,B,B,B,A,A} could not ever occur? This looks like a valid sequence of wins to me... – Code-Guru Jul 18 '12 at 19:22
Once team B wins 4 games the series is over. – crf Jul 18 '12 at 19:24
See David Spencer's answer below. – Code-Guru Jul 18 '12 at 19:32
To make more money, the league has decided that the series will go $7$ games, but the usual rules for determining the winner (first to win $4$) apply. Then Team B wins the modified series iff it wins the real series. – André Nicolas Jul 18 '12 at 20:54
This is not a coincidence, you are looking at the same problem in two different ways. The situation {B,A,B,B,B,A,A} is irrelevant in your friend's argument, because you are not counting the number of different possible sequences of events.
Rather, you are attempting to estimate the probability of at least $n$ successes in $k$ trials, which is exactly what the binomial distribution does.
In your second approach, you are essentially estimating the number of possible legal sequences terminating in a victory condition, and then computing the total fraction out of all possible sequences.
Both approaches are equivalent. One is a little easier to implement ;)
-
He is correct. Although this does include series that would be over after fewer than $7$ games "in real life", they would still all result in team B winning, no matter what happens after team B gets $4$ wins.
In your example, the probability that one of
{B,A,B,B,B,A,A}
{B,A,B,B,B,A,B}
{B,A,B,B,B,B,A}
{B,A,B,B,B,B,B}
happening without terminating series after four wins is equivalent to the probability of {B,A,B,B,B} happening across the first five games because later events do not affect anything.
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How to find the argument of $\cos 2 - i\sin 2$
How would you find the argument of the following number. $$\cos{2}-i\sin{2}$$
I'm aware that complex numbers in the form $r(\cos{\theta}+i\sin{\theta})$ have an argument of $\theta$, but what do you do with the $-$ sign?
Regardless of method, you will demonstrate some knowledge of Euler's identity and/or trigonometric identities.
Method I: Arctangent of slope
The argument is the arctangent of the ratio of the imaginary component to the real component, accounting for quadrant. If you forget to account for quadrant, this gives arguments in the interval $(-\pi/2, \pi/2)$, "half" of which are wrong. (Recall that arctangent is not a single-valued inverse. This is obvious when one remembers that tangent is periodic.)
Since $\cos 2 < 0$ and $-\sin 2 < 0$, this is in quadrant III.
\begin{align*} \arctan \frac{-\sin 2}{\cos 2} &= \arctan(-\tan 2)) \\ &= - \arctan \tan 2 &&\text{$\arctan$ is an odd function}\\ &= - \arctan \tan (2 - \pi) &&\text{$\tan$ has period $\pi$}\\ &= -(2 - \pi +\pi k), k \in \mathbb{Z} &&-\pi/2 < 2 - \pi < \pi/2 \text{.} \end{align*} Choosing $k=1$, this lands in quadrant III, giving the argument $-2$. If we had just mechanically evaluated, say with a calculator, \begin{align*} \arctan \frac{-\sin 2}{\cos 2} &= \arctan(2.18504\dots) \\ &= 1.14159\dots{} + \pi k, k \in \mathbb{Z} \text{.} \end{align*} The subset of these in quadrant III have $k$ odd. Picking $k = -1$, we get $-2$.
Of all the $k$ that give a result in the correct quadrant, which do you pick? You pick the one that conforms to your conventions for the range of value of an argument, if you have such a convention. If you do not, pick the one that makes your subsequent steps easier (or don't pick and leave the result as an equivalence class $\mod 2 \pi$).
Method II: Conjugation | {
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Method II: Conjugation
If your only problem is the wrong sign of imaginary component, use conjugation. \begin{align*} \arg(\cos 2 - \mathrm{i} \sin 2) &= \arg(\overline{\cos 2 + \mathrm{i} \sin 2}) \\ &= \arg(\overline{\mathrm{e}^{\mathrm{i} 2}}) \\ &= -\arg(\mathrm{e}^{\mathrm{i} 2}) \\ &= -2 \text{.} \end{align*}
Method III: Even-odd identities
Sine is odd. Cosine is even. This is expressed in the even-odd identities. So $\cos(-2) = \cos(2)$ and $\sin(-2) = -\sin(2)$. Consequently, \begin{align*} \cos 2 - \mathrm{i} \sin 2 &= \cos -2 + \mathrm{i} \sin -2 \\ &= \mathrm{e}^{\mathrm{i}(-2)} \text{,} \end{align*} having argument $-2$.
Brahadeesh's and MrYouMath's answers use this method, without identifying what was done. Michael Rozenberg's answer combines this with the periodicity (by $2\pi$) identity, again without identifying what was done.
• Thank you very much. This has really cleared things up for me! – Ewan Miller Sep 9 '17 at 20:53
You can write this complex number as $\cos(-2) + i \sin(-2)$. Now you can see that the argument is $-2$.
$z=r(\sin\theta+i\sin\theta)$, where $r\geq0$ and $\theta=\arg{z}\in[0,2\pi)$.
$$\cos2-i\sin2=\cos(2\pi-2)+i\sin(2\pi-2).$$
Thus, $\arg(\cos2-i\sin2)=2\pi-2$. | {
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$$\cos2-i\sin2=\cos(2\pi-2)+i\sin(2\pi-2).$$
Thus, $\arg(\cos2-i\sin2)=2\pi-2$.
• Thank you. Am I right in thinking then that there is a general rule that r(cosA-isinA)=r(cos(2π-A)+isin(2π-A))? – Ewan Miller Sep 9 '17 at 16:05
• @Ewan Miller It depends on definition. Sometimes $\arg{z}\in[0,2\pi)$ and sometimes $\arg{z}\in[-\pi,\pi)$. You need to ask your teacher. I like the first versa. – Michael Rozenberg Sep 9 '17 at 16:13
• In the case where $\arg{z}\in[-\pi,\pi)$, would it be r(cosA-isinA)=r(cos(-A)+isini(-A))? – Ewan Miller Sep 9 '17 at 16:17
• Also, I haven't got a teacher as I have just left school and am teaching this to myself in my gap year – Ewan Miller Sep 9 '17 at 16:19
• @Ewan Miller It's not so easy. If $A\in(-\pi,\pi)$ then you are right:$r(\cos(-A)+i\sin(-A))$ , but for $A=-\pi$ we get $r(\cos{A}+i\sin{A})$. All this depend on the level of your teacher. Maybe he don't see here problems. – Michael Rozenberg Sep 9 '17 at 16:23 | {
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# Thread: Counting, Permutation, and Combinations
1. ## Counting, Permutation, and Combinations
Guys I have several questions that I would like to ask about:
1. In how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, if the bride and the groom are among these 10 people, if:
a. the bride must be in the picture
b. exactly one of the bride and the groom is in the picture
2. How many ways are there for 10 women and six men to stand in a line so that no two men stand next to each other?
3. A professor writes 40 discrete mathematics true/false questions. Of these statements in these questions, 17 are true. If the questions can be positioned in any order, how many different answer keys are possible.
1a. because the bride is already included among 6 people, therefore there are 5 more people that we can choose and arrange from, so it's 5!. I am not sure if this is right
1b. I don't quite understand the question
2. P(10,10) x P(11,6)
3. C(40,17)
I am not quite sure of all these answers, but suggestions/help to my questions are highly appreciated
2. a. the bride must be in the picture
Let us divide this problem into cases.
Code:
BXXXXX
XBXXXX
XXBXXX
XXXBXX
XXXXBX
XXXXXB
Where "X" stands for any other person. And "B" stands for bride. We will find the number of such cases and then add all of them up together.
In the first case there are $\displaystyle (1)(9)(8)(7)(6)(5) = 15120$. Similarly in each case we have the same number of arrangements. So in total we have $\displaystyle 6\cdot 15120 = 90720$.
b. exactly one of the bride and the groom is in the picture
Again divide this into cases. Case 1 is that bride is in picture and groom is not. Case 2 is that bride is not in picture and groom is. Then add those results together. Each subcase is similar to the problem just did above. | {
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3. Originally Posted by ThePerfectHacker
Let us divide this problem into cases.
Code:
BXXXXX
XBXXXX
XXBXXX
XXXBXX
XXXXBX
XXXXXB
Where "X" stands for any other person. And "B" stands for bride. We will find the number of such cases and then add all of them up together.
In the first case there are $\displaystyle (1)(9)(8)(7)(6)(5) = 15120$. Similarly in each case we have the same number of arrangements. So in total we have $\displaystyle 6\cdot 15120 = 90720$.
Again divide this into cases. Case 1 is that bride is in picture and groom is not. Case 2 is that bride is not in picture and groom is. Then add those results together. Each subcase is similar to the problem just did above.
as for case 1:
the bride is in the picture but the groom is not, 1 x 8 x 7 x 6 x 5 x 4 = 6,720. 6720 x 6 = 13,440. I choose 8 because therefore the number of possibilities of people chosen is now 9 instead of 10 because the groom is excluded here.
and for case 2:
the bride is not in picture but the groom is has the same possibilities as case 1 right?
then we just add case 1 and case 2?
4. Originally Posted by TheRekz
as for case 1:
the bride is in the picture but the groom is not, 1 x 8 x 7 x 6 x 5 x 4 = 6,720. 6720 x 6 = 13,440. I choose 8 because therefore the number of possibilities of people chosen is now 9 instead of 10 because the groom is excluded here.
and for case 2:
the bride is not in picture but the groom is has the same possibilities as case 1 right?
then we just add case 1 and case 2?
I agree with you.
5. what if now the question is both the bride and the groom must be in the picture:
(1 x 1 x 8 x 7 x 6 x 5) = 1,680 x 6 = 10,080
is this right?
6. Hello, TheRekz!
1. In how many ways can a photographer at a wedding
arrange 6 people in a row from a group of 10 people,
if the bride and the groom are among these 10 people, and:
a. the bride must be in the picture
b. exactly one of the bride and the groom is in the picture | {
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a) The bride is in the picture.
Select five more from the other nine people: .$\displaystyle C(9,5)$ ways.
These six people can be arranged in $\displaystyle 6!$ ways.
Therefore, there are: .$\displaystyle C(9,5) \times 6!$ ways.
b) Either bride or the groom is in the picture (but not both).
There are 2 choices for the newlywed to be in the picture.
The other 5 people are chosen from the remaining 8 people (not the other newlywed).
. . There are: .$\displaystyle C(8,5)$ ways.
The six people can be arranged in $\displaystyle 6!$ ways.
Therefore, there are: .$\displaystyle 2 \times C(8,5) \times 6!$ ways.
2. How many ways are there for 10 women and six men to stand in a line
so that no two men stand next to each other?
Arrange the tenwomen in a row. .There are $\displaystyle P(10,10) = 10!$ possible arrangements.
Leave spaces between the women: .$\displaystyle \_\;W\;\_\;W\;\_\;W\;\_\;W\;\_\;W\;\_\;W\;\_\;W\;\ _\;W\;\_\;W\;\_\;W\;\_$
Arrange the six men in any six of the eveleven spaces.
. . There are: .$\displaystyle P(11,6)$ ways.
Therefore, there are: .$\displaystyle P(10,10) \times P(11,6)$ ways.
3. A professor writes 40 discrete mathematics true/false questions.
Of these statements in these questions, 17 are true.
If the questions can be positioned in any order,
how many different answer keys are possible?
Your answer is correct . . . $\displaystyle C(40,17)$
7. Originally Posted by TheRekz
what if now the question is both the bride and the groom must be in the picture:
(1 x 1 x 8 x 7 x 6 x 5) = 1,680 x 6 = 10,080
is this right?
If both the bride and groom are in the picture then:
fix the bride in the first position:
BGXXXX = 1*1*8*7*6*5
BXGXXX = 1*8*1*7*6*5
.
.
.
BXXXXG = 1*8*7*6*5*1
Then there are 5*(1*1*8*7*6*5) ways to arrange them with the bride in the first position. | {
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Then there are 5*(1*1*8*7*6*5) ways to arrange them with the bride in the first position.
Now repeat with the bride in the second, third,....,sixth position and you get the same result each time. Therefore, if you add these all up you get (5*(1*1*8*7*6*5))*6 = 50,400
,
,
,
# A professor writes 40 discrete mathematics true/false
Click on a term to search for related topics. | {
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# Integration by Trigonometric Substitution vs Table of Integral Solution
I'm not sure how to phrase this question, but I find myself confused over the correctness of a particular solution in the table of integral, specifically: $$\int\frac{dx}{\sqrt{x^2-a^2}}=\ln|\sqrt{x^2-a^2}+x|+C$$
I find that if you try to solve it with trigonometric substitution, you get a different answer:
So, for the example, since the form $$\sqrt{x^2-a^2}$$ is present, we use the substitution $$x=asec \theta$$, and $$dx=asec\theta tan \theta d\theta$$. Doing so:
$$\int\frac{asec\theta tan \theta d\theta}{\sqrt{(asec \theta)^2-a^2}}$$ $$\int\frac{asec\theta tan \theta d\theta}{\sqrt{a^2sec^2 \theta-a^2}}$$ $$\int\frac{asec\theta tan \theta d\theta}{a\sqrt{sec^2 \theta-1}}$$ Using the trigonometric identity $$tan^2 \theta=sec^2 \theta -1$$ $$\int\frac{asec\theta tan \theta d\theta}{atan \theta}$$
Cancelling like terms, we get $$\int sec \theta$$, which finally integrates to $$ln|sec \theta + tan \theta|+C$$
Undoing the substitution, using this triangle:
https://i.stack.imgur.com/6tkR7.png
With $$sec \theta = \frac{x}{a}$$ and $$tan \theta = \frac{\sqrt{x^2-a^2}}{a}$$, the final answer then is:
$$ln\biggl|\frac{x}{a} + \frac{\sqrt{x^2-a^2}}{a}\biggl|+C$$
Which is evidently different from the one from the table of integrals. Is my solution wrong or something?
• looks the same to me, since $-\ln |a|$ can be absorbed into $C$? – Calvin Khor Feb 7 at 13:48
• Oh right, that can be done. Thanks for pointing that out – Xyzar Feb 7 at 13:52
$$\ln \left|\dfrac{x+\sqrt{x^2-a^2}}{a}\right| +c= \ln |x+\sqrt{x^2-a^2}| - \ln|a| +c = \ln(x+\sqrt{x^2-a^2}) +k$$
$$k = c-\ln |a| =$$ constant | {
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# Different Ways of Integrating $3\sin x\cos x$
I am asking this question for my son who is in (equivalent) twelfth grade and I failed to answer his query.
When he tries to integrate $3\sin x\cos x$, he finds that this can be done in at least following three ways. And these three ways do not produce equivalent results.
ONE
Let us assume, $\sin x = z$.
This gives, \begin{align*} \cos x &= \frac{dz}{dx}\\ \cos x dx &= dz \end{align*}
So, we can write,
\begin{align*} \int 3\sin x\cos x dx &=3 \int zdz\\ &=3 \frac{z^2}{2}\\ &=\frac{3}{2} \sin^2 x\\ &=\frac{3}{4}\times 2\sin^2 x\\ &=\frac{3}{4} (1 -\cos 2x)\\ \end{align*}
TWO
Let us assume, $\cos x = z$.
This gives, \begin{align*} -\sin x &= \frac{dz}{dx}\\ \sin x dx &= -dz \end{align*}
So, we can write,
\begin{align*} \int 3\sin x\cos x dx &=-3 \int zdz\\ &=-3 \frac{z^2}{2}\\ &=-\frac{3}{2} \cos^2 x\\ &=-\frac{3}{4}\times 2\cos^2 x\\ &=-\frac{3}{4} (1 +\cos 2x)\\ \end{align*}
THREE
\begin{align*} \int 3\sin x\cos x dx &=\frac{3}{2}\int 2\sin x\cos x dx\\ &=\frac{3}{2}\int \sin 2x dx\\ &=-\frac{3}{2}\times\frac{1}{2} \cos 2x\\ &=-\frac{3}{4} \cos 2x\\ \end{align*}
The results found in above three methods are not the same.
If we try a simple approach of evaluating the integration results at, $x = \frac{\pi}{6}$, we get as follows.
From the first one,
$\frac{3}{4} (1 -\cos 2x) = \frac{3}{4} (1 -\cos \frac{2\pi}{6}) = \frac{3}{4} (1 -\cos \frac{\pi}{3}) = \frac{3}{4} (1 - \frac{1}{2}) = \frac{3}{4}\times\frac{1}{2} = \frac{3}{8}$
From the second one,
$-\frac{3}{4} (1 +\cos 2x) = -\frac{3}{4} (1 +\cos \frac{2\pi}{6}) = -\frac{3}{4} (1 +\cos \frac{\pi}{3}) = -\frac{3}{4} (1 + \frac{1}{2}) = -\frac{3}{4}\times\frac{3}{2} = -\frac{9}{8}$
From the third one,
$-\frac{3}{4} \cos 2x=-\frac{3}{4} \cos \frac{2\pi}{6} = -\frac{3}{4} \cos \frac{\pi}{3} = -\frac{3}{4} \times \frac{1}{2} = -\frac{3}{8}$ | {
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Clearly, we are getting some nonequivalent results. We have failed to find the mistakes or explanations behind this. Your help will be appreciated.
-
You can locate an several primitives of a function, they differ in a constant, but that does not mean that the results are wrong. – Hiperion Sep 16 '13 at 5:50
Indefinite integration can be 'misleading' in this sense. – copper.hat Sep 16 '13 at 5:52
A great question to ask students to see if they can come up with Zev's answer (or equivalent). – nbubis Sep 16 '13 at 6:01
– Jonas Meyer Sep 16 '13 at 6:07
When I was doing A-levels, I came across a similar example, but the two "different" answers to the integral involved respectively $\tan^2(x)$ and $\sec^2(x)$. Of course, these two expressions differ by a constant, but the explanation was non-obvious to me (and my dad) then, and I had to get my teacher to explain it. – Hammerite Sep 16 '13 at 9:29
You're forgetting that an indefinite integral must include a constant of integration; for any chosen constant $C$, we have that $$\frac{d}{dx}\left(-\frac{3}{4}\cos(2x)+C\right)=3\sin(x)\cos(x),$$ and that is precisely the relationship captured by the statement that
$$\int 3\sin(x)\cos(x)\,dx=-\frac{3}{4}\cos(2x)+C.$$ | {
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$$\int 3\sin(x)\cos(x)\,dx=-\frac{3}{4}\cos(2x)+C.$$
-
Indeed this is a great demonstration of that's so important. – David H Sep 16 '13 at 5:50
So, if is asked to solve the problem in the examination, which approach he should follow? His textbook contains only one of the answers. And as you can see, some graders may like only the conventional answers. No disrespect intended. – Masroor Sep 16 '13 at 6:01
@MMA - All the answers are equally "wrong" since non of them contain a constant. When you let the constant C "swallow" the other constant term, all three give the same result. – nbubis Sep 16 '13 at 6:02
@nbubis But his text book contains only one of the answers, (I can not get the book now), and given the prevailing circumstances, it will be difficult to modify the system. I can put down the answer which is the book in six/seven hours when the book comes back from school. – Masroor Sep 16 '13 at 6:07
@MMA I sympathize with your situation, especially if your particular school has you feeling that cynical towards the system. But I have no way of predicting what which technique a grader would consider most appropriate, because there isn't one (and if the grader doesn't know it then your son knows more calculus than his teacher). He should learn all three methods. Solving integrals is like choosing a golf-club for a given shot. The best one is the one that gets the job done. – David H Sep 16 '13 at 6:23
All three answers are correct provided you add a constant to each one of those.
Because from the very definition of integration, it is the area under the curve, so it requires bounds to give a unique value. You can't evaluate the value of an indefinite integral without including constant.
And I am sure that in the examination, your son won't be asked to evaluate the value of an integral without providing limits of integration or providing its value at some other point. | {
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For instance, in question it may be mentioned that evaluate the value of expression at x=π/6 , given its value at x=0 is 1. So in this case, all three answers will give the correct value i.e. 11/8
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# [SOLVED]chain rule and e^(u)
#### DeusAbscondus
##### Active member
Hi folks,
I don't know if my experience is at all common (and I would like some feedback on this if possible), but I can't seem to nail down the properties of euler's number in the context of chain rule problems.
Here is the nub of my difficulty:
1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$
This I accept, though, not having seen a formal proof of it, and since it is counter-intuitive, I must take it on faith.
But the following I do not understand; could someone help me towards understanding?
2. $\text{If }g(x)=e^{x^{4}} \text{then }g'(x)=4e^{x^4}x^3$
Am I on the right track to observe that, if 2. is correct, then g(x) is a composite function, hence subject to the chain rule?
And if so, is the following a generally valid way to work through this and all such problems:
$\text{If }g(x)=e^{x^{4}} \text{find }g'(x)$
$\text{Now}g'(x)=u'v' \text{by Chain Rule}$
$\text{So, let }u=x^4 \text{and }v=e^u$
$\text{Then }u'=4x^3 \text{and }v'=e^u \text{ (by some rule which currently exceeds my understanding)}$
$\text{Therefore }g'(x)=u'v'=4x^3*e^u=\text{(via substitution) }4x^3*e^{x^{4}}$
$\text{Which, simplified }=4e^{x^{4}}x^3$
Finally, I have a similar hesitation/scruple/sense of vertigo when it comes to dealing with another unusual derivative, that of:
$ln(x)$
If anyone can see why, having the read the foregoing, I would feel unsure of myself around this animal, could they possibly add some notes to help me tame it?
Regs,
DeusAbs
Last edited:
#### Jameson
Staff member
Hi DeusAbscondus,
1) Let's try to derive the derivative of $e^x$. Recall that for $f(x)$ it follows that $$\displaystyle f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$.
In this case that becomes $$\displaystyle \lim_{h \rightarrow 0} \frac{e^{x+h}-e^x}{h}=\frac{e^x \cdot e^h-e^x}{h}=\frac{e^x(e^h-1)}{h}$$. | {
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Ok, from here we should notice that $e^x$ isn't dependent on h in this limit thus can be brought outside of the computation as long as we multiply through at the end.
So the last bit becomes $$\displaystyle \left( e^x \cdot \lim_{h \rightarrow 0} \frac{e^h-1}{h} \right)$$.
From here you need to compute that limit and you can do it a few ways. For a less rigorous approach that is more intuitive as well, I suggest graphing $$\displaystyle \frac{e^h-1}{h}$$ on a graphing calculator and you'll see that as h approaches zero, then the y value approaches 1.
So at the end we get that $$\displaystyle \left( e^x \cdot \lim_{h \rightarrow 0} \frac{e^h-1}{h} \right) = e^x \cdot (1) = e^x$$, therefore $$\displaystyle \frac{d}{dx} e^x = e^x$$
Please let me know if any of that is unclear. Again, it's not a rigorous proof by any means, more of a "walk through".
2) You are completely correct that $$\displaystyle \frac{d}{dx} e^{f(x)} = e^{f(x)} \cdot f'(x)$$. As you said this is an application of the chain rule.
#### Sudharaka
##### Well-known member
MHB Math Helper
$\text{Then }u'=4x^3 \text{and }v'=e^u \text{ (by some rule which currently exceeds my understanding)}$
Hi DeusAbscondus,
The only rule that is used here is the chain rule.
\begin{eqnarray}
g(x)&=&e^{x^{4}}\\
g'(x)&=&\frac{d}{d(x^4)}e^{x^{4}}\frac{d}{dx}x^4\\
&=&e^{x^{4}}(4x^3)\\
\therefore g'(x)&=&4x^3e^{x^{4}}
\end{eqnarray}
Kind Regards,
Sudharaka.
#### Fantini
##### "Read Euler, read Euler." - Laplace
MHB Math Helper
Perhaps to avoid particularizing and help you think of the exponential as any other function, think of $e^{x^4}$ as $h(x) = e^x$ and $g(x) = x^4$, then $(h \circ g)(x) = h(g(x)) = e^{x^4}$ and $(h \circ g)' (x) = h'(g(x)) \cdot g'(x) = e^{x^4} \cdot (4x^3) = 4 e^{x^4} x^3.$
#### chisigma
##### Well-known member
Hi DeusAbscondus,
1) Let's try to derive the derivative of $e^x$. Recall that for $f(x)$ it follows that $$\displaystyle f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$. | {
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In this case that becomes $$\displaystyle \lim_{h \rightarrow 0} \frac{e^{x+h}-e^x}{h}=\frac{e^x \cdot e^h-e^x}{h}=\frac{e^x(e^h-1)}{h}$$.
Ok, from here we should notice that $e^x$ isn't dependent on h in this limit thus can be brought outside of the computation as long as we multiply through at the end.
So the last bit becomes $$\displaystyle \left( e^x \cdot \lim_{h \rightarrow 0} \frac{e^h-1}{h} \right)$$.
From here you need to compute that limit and you can do it a few ways...
One possible 'rigorous' way to demonstrate that...
$\displaystyle \lim_{h \rightarrow 0} \frac{e^{h}-1}{h}=1$ (1)
... is the following...
a) You start from the definition of exponential...
$\displaystyle e^{h}= \lim_{n \rightarrow \infty} (1+\frac{h}{n})^{n}$ (2)
b) You demonstrate the identity...
$\displaystyle \lim_{n \rightarrow \infty} (1+\frac{h}{n})^{n}= \lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac {h^{k}}{k!} = 1 + h + \frac {h^{2}}{2}+... \frac{h^{n}}{n!}+...$ (3)
c) from (3) You derive...
$\displaystyle \frac{e^{h}-1}{h}= 1 + \frac{h}{2} +... + \frac{h^{n-1}}{n!}+...$ (4)
... and the (1) follows immediately...
Kind regards
$\chi$ $\sigma$
#### DeusAbscondus
##### Active member
thanks: Jameson, Sudharaka and Fantini: Jameson for the informal proof of $e^x$, Sudharaka for reminding me that there is no mystery here, just the chain rule and Fantini for the code behind $(h \circ g)$ and some other stuff besides.
Just a quick follow up:
can one of you gents refer me to a gentle approach to understanding a delta/epsilon proof, via video preferably, so as I can see it being worked out, pause it and study it?
and, Jameson, is this proof: the delta/epsilon proof, the more formal proof of the first derivative to which your notes allude?
thx again gentelmen,
DeusAbs
#### Jameson | {
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thx again gentelmen,
DeusAbs
#### Jameson
Staff member
It's not actually completely informal the way I did it, just not rigorous. By applying chisigma's technique as well as many other possibilities, such as L'Hopital's Rule you can soundly demonstrate that this limit is 1. A delta-epsilon proof of this limit wouldn't be asked in basic calculus course as far as I know.
Anyway, as to your question. I'm not sure about videos but here's a link that's part of a great Calculus help site I would recommend for anyone, Paul's Online Notes. The link I gave you specifically deals with delta-epsilon proofs and has nice diagrams plus examples that cover both the theory and show you how it's done in practice.
Let me know what you think!
#### Sudharaka
##### Well-known member
MHB Math Helper
thanks: Jameson, Sudharaka and Fantini: Jameson for the informal proof of $e^x$, Sudharaka for reminding me that there is no mystery here, just the chain rule and Fantini for the code behind $(h \circ g)$ and some other stuff besides.
Just a quick follow up:
can one of you gents refer me to a gentle approach to understanding a delta/epsilon proof, via video preferably, so as I can see it being worked out, pause it and study it?
and, Jameson, is this proof: the delta/epsilon proof, the more formal proof of the first derivative to which your notes allude?
thx again gentelmen,
DeusAbs
Hi DeusAbscondus,
A good place filled with video lessons is Khan Academy. Here is a link to the Epsilon Delta limit definition introduction video.
Kind Regards,
Sudharaka.
#### CaptainBlack
##### Well-known member
Hi folks,
I don't know if my experience is at all common (and I would like some feedback on this if possible), but I can't seem to nail down the properties of euler's number in the context of chain rule problems.
Here is the nub of my difficulty:
1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$ | {
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Here is the nub of my difficulty:
1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$
This I accept, though, not having seen a formal proof of it, and since it is counter-intuitive, I must take it on faith.
But the following I do not understand; could someone help me towards understanding?
From the way this is worded it seems that you think that there is some universally agreed definition of the exponential function. The truth is that there is not, you may have been told it is defined as:
$\exp(x)=\lim_{n \to \infty} \left(1+\frac{x}{n}\right)^n$
but it is often defined as the solution to the differential equation initial value problem: $$f'(x)=f(x), f(0)=1$$, or as the inverse of the natural logarithm, ..
So to prove that the derivative of the exponential function is itself we really need to know what definition you are starting from (and also what you already know).
CB
#### DeusAbscondus
##### Active member
From the way this is worded it seems that you think that there is some universally agreed definition of the exponential function. The truth is that there is not, you may have been told it is defined as:
$\exp(x)=\lim_{n \to \infty} \left(1+\frac{x}{n}\right)^n$
but it is often defined as the solution to the differential equation initial value problem: $$f'(x)=f(x), f(0)=1$$, or as the inverse of the natural logarithm, ..
So to prove that the derivative of the exponential function is itself we really need to know what definition you are starting from (and also what you already know).
CB
I just had a look at graphs of these functions: given that they are mirror images of each other graphically, I would have thought their inter-definability (if i can put it thus) bordered the trivial. | {
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I'd also have thought my post here (indeed, the combined oeuvre of my posts to date ) makes perfectly clear my starting position in the knowledge of mathematics stakes: zero + epsilon.
Prescinding from that, any light you can throw on it would seem to me to be, well - in a dark place - illuminating, n'est-ce pas?
I always enjoy your sparse, ascetical take on things, Cap'n.
So, thanks for the input,
Deus Abs
#### DeusAbscondus
##### Active member
It's not actually completely informal the way I did it, just not rigorous. By applying chisigma's technique as well as many other possibilities, such as L'Hopital's Rule you can soundly demonstrate that this limit is 1. A delta-epsilon proof of this limit wouldn't be asked in basic calculus course as far as I know.
Anyway, as to your question. I'm not sure about videos but here's a link that's part of a great Calculus help site I would recommend for anyone, Paul's Online Notes. The link I gave you specifically deals with delta-epsilon proofs and has nice diagrams plus examples that cover both the theory and show you how it's done in practice.
Let me know what you think!
Thanks Jameson; just spent some hours there! I suffer from a dearth of really first-rate material and my course notes are a nest of errors and poorly edited material, so I appreciate such a rich fund of examples, proofs, worked examples and tips.
I'll be sure to make a fuller response once I've had more time to check it.
DeusAbs
#### chisigma
##### Well-known member
From the way this is worded it seems that you think that there is some universally agreed definition of the exponential function. The truth is that there is not, you may have been told it is defined as:
$\exp(x)=\lim_{n \to \infty} \left(1+\frac{x}{n}\right)^n$
but it is often defined as the solution to the differential equation initial value problem: $$f'(x)=f(x), f(0)=1$$, or as the inverse of the natural logarithm, .. | {
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So to prove that the derivative of the exponential function is itself we really need to know what definition you are starting from (and also what you already know).
CB
More that 280 years ago [!] Leonhard Euler defined in such way the exponential and natural log functions...
$\displaystyle e^{z}= \lim_{n \rightarrow \infty} (1+\frac{z}{n})^{n}$
$\displaystyle \ln z= \lim_{n \rightarrow \infty} n\ (z^{\frac{1}{n}}-1)$ (1)
... and also demonstrated that the two function are inverse to one other. In my [humble ...] opinion the definitions (1) are also today 'the best' mainly because they are valid 'without additions' for any real or complex value of z...
Kind regards
$\chi$ $\sigma$
#### CaptainBlack
##### Well-known member
I'd also have thought my post here (indeed, the combined oeuvre of my posts to date ) makes perfectly clear my starting position in the knowledge of mathematics stakes: zero + epsilon.
Prescinding from that, any light you can throw on it would seem to me to be, well - in a dark place - illuminating, n'est-ce pas?
You must have at least some idea of how the exponential function is defined, or at the very least how "e" is defined, otherwise you are starting in the middle.
(I would recommend you get a copy of Morris Kline's book "Calculus; An intuitive and physical approach", the only flaw of which is its use of US Customary Units)
CB
#### Opalg
##### MHB Oldtimer
Staff member
More than 280 years ago [!] Leonhard Euler defined in such way the exponential and natural log functions...
$\displaystyle e^{z}= \lim_{n \rightarrow \infty} \Bigl(1+\frac{z}{n}\Bigr)^{n}$
$\displaystyle \ln z= \lim_{n \rightarrow \infty} n\ (z^{\frac{1}{n}}-1)$ (1) | {
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$\displaystyle \ln z= \lim_{n \rightarrow \infty} n\ (z^{\frac{1}{n}}-1)$ (1)
... and also demonstrated that the two function are inverse to one other. In my [humble ...] opinion the definitions (1) are also today 'the best' mainly because they are valid 'without additions' for any real or complex value of z...
In my [equally humble ...] opinion, the definitions (1) are not the best, because they are hard to work with. For example, starting from (1) how would you prove the index law $e^{x+y} = e^xe^y$, or the derivative rule $\frac d{dx}e^x = e^x$? Those proofs can surely be given, but the limit definition does not strike me as a natural or convenient one to start from. I much prefer the power series definition $e^x = \sum_{n=0}^\infty\frac{x^n}{n!}$ for the (complex) exponential. The basic properties follow quite easily from that, as does the fact that the real exponential $e^x:\mathbb{R}\to(0,\infty)$ increases monotonically and hence has an inverse, which defines the real logarithm function. (The complex logarithm is not a single-valued function, and must therefore be defined more indirectly.)
#### chisigma | {
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#### chisigma
##### Well-known member
In my [equally humble ...] opinion, the definitions (1) are not the best, because they are hard to work with. For example, starting from (1) how would you prove the index law $e^{x+y} = e^xe^y$, or the derivative rule $\frac d{dx}e^x = e^x$? Those proofs can surely be given, but the limit definition does not strike me as a natural or convenient one to start from. I much prefer the power series definition $e^x = \sum_{n=0}^\infty\frac{x^n}{n!}$ for the (complex) exponential. The basic properties follow quite easily from that, as does the fact that the real exponential $e^x:\mathbb{R}\to(0,\infty)$ increases monotonically and hence has an inverse, which defines the real logarithm function. (The complex logarithm is not a single-valued function, and must therefore be defined more indirectly.)
On the basis of the [easily enough demonstrable...] identity...
$\displaystyle \lim_{n \rightarrow \infty} (1+\frac{z}{n})^{n} = \lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{z^{k}}{k!}$ (1)
... the two definitions must be considered fully equivalent. On the purely conceptual basis however the original Euler's definition requires only the concept of limit and is 'more elementary' than any other definition and that is an excellent reason for defining it as 'the best'... at least for me..
Kind regards
$\chi$ $\sigma$
#### QuestForInsight
##### Member
1. $\text{If }f(x)=e^x \text{then }f'(x)=e^x$
This is very easy to get using the power series definition from Opalg's post.
\begin{aligned} & (e^x)' = \bigg(\sum_{k \ge 0}\frac{x^k}{k!}\bigg)' = \sum_{k \ge 0}\frac{kx^{k-1}}{k!} = \sum_{k \ge 1}\frac{kx^{k-1}}{k!} = \sum_{k+1 \ge 1}\frac{(k+1)x^{k}}{(k+1)!} = \sum_{k \ge 0}\frac{x^{k}}{k!}.\end{aligned}
#### chisigma
##### Well-known member
This is very easy to get using the power series definition from Opalg's post. | {
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\begin{aligned} & (e^x)' = \bigg(\sum_{k \ge 0}\frac{x^k}{k!}\bigg)' = \sum_{k \ge 0}\frac{kx^{k-1}}{k!} = \sum_{k \ge 1}\frac{kx^{k-1}}{k!} = \sum_{k+1 \ge 1}\frac{(k+1)x^{k}}{(k+1)!} = \sum_{k \ge 0}\frac{x^{k}}{k!}.\end{aligned}
That is true of course but [conceptually...] the preliminary demonstration of the convergence theorem of the series of derivatives is required. The computation of derivative of $e^{x}$ using the standard derivative definition require only to demonstrate that...
$\displaystyle \lim_{h \rightarrow 0} \frac {e^{h}-1}{h}=1$ (1)
... and that requires elementary instruments...
Kind regards
$\chi$ $\sigma$ | {
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# 77 to use the alternating series test consider a n f
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Chapter 11 / Exercise 77
Single Variable Calculus
Stewart
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77. To use the alternating series test, consider a n = f ( n ) , where f ( x ) = arctan( a/x ) . We need to show that f ( x ) is decreasing. Since f 0 ( x ) = 1 1 + ( a/x ) 2 - a x 2 , we have f 0 ( x ) < 0 for a > 0 , so f ( x ) is decreasing for all x . Thus a n +1 < a n for all n , and as lim n →∞ arctan( a/n ) = 0 for all a , by the alternating series test, X n =1 ( - 1) n arctan( a/n ) converges. 78. The n th partial sum of the series is given by S n = 1 - 1 2 + 1 3 - · · · + ( - 1) n - 1 n , so the absolute value of the first term omitted is 1 / ( n + 1) . By Theorem 9.9, we know that the value, S , of the sum differs from S n by less than 1 / ( n +1) . Thus, we want to choose n large enough so that 1 / ( n +1) 0 . 01 . Solving this inequality for n yields n 99 , so we take 99 or more terms in our partial sum. 79. The n th partial sum of the series is given by S n = 1 - 2 3 + 4 9 - · · · + ( - 1) ( n - 1) 2 3 ( n - 1) ,
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750 Chapter Nine /SOLUTIONS so the absolute value of the first term omitted is (2 / 3) n . By Theorem 9.9, we know that the value, S , of the sum differs from S n by less than (2 / 3) n . Thus, we want to choose n large enough so that (2 / 3) n 0 . 01 . Solving this inequality for n yields n 11 . 358 , so taking 12 or more terms in our partial sum is guaranteed to be within 0 . 01 of the sum of the series. Note: Since this is a geometric series, we know the exact sum to be 1 / (1+2 / 3) = 0 . 6 . The partial sum S 12 is 0 . 595 , which is indeed within 0 . 01 of the sum of the series. Note, however, that S 11 = 0 . 6069 , which is also within 0 . 01 of the exact sum of the series. Theorem 9.9 gives us a value of n for which S n is guaranteed to be within a small tolerance of the sum of an alternating series, but not necessarily the smallest such value. 80. The n th partial sum of the series is given by S n = 1 2 - 1 24 + 1 720 - · · · + ( - 1) n - 1 (2 n )! , so the absolute value of the first term omitted is 1 / (2 n + 2)! . By Theorem 9.9, we know that the value, S , of the sum differs from S n by less than 1 / (2 n +2)! . Thus, we want to choose n large enough so that 1 / (2 n +2)! 0 . 01 . Substituting n = 2 into the expression 1 / (2 n + 2)! yields 1 / 720 which is less than 0.01. We therefore take 2 or more terms in our partial sum. 81. Since 0 c n 2 - n for all n , and since 2 - n is a convergent geometric series, c n converges by the Comparison Test. Similarly, since 2 n a n , and since 2 n is a divergent geometric series, a n diverges by the Comparison Test. We do not have enough information to determine whether or not b n and d n converge. 82. (a) The sum a n · b n = 1 /n 5 , which converges, as a p -series with p = 5 , or by the integral test: Z 1 1 x 5 dx = lim b →∞ x - 4 ( - 4) b 1 = lim b →∞ b - 4 ( - 4) + 1 4 = 1 4 . Since this improper integral converges, a n · b n also converges. (b) This is an alternating series that satisfies the conditions of the | {
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a n · b n also converges. (b) This is an alternating series that satisfies the conditions of the alternating series test: the terms are decreasing and have limit 0 , so ( - 1) n / n converges. (c) We have a n b n = 1 /n , so a n b n is the harmonic series, which diverges. | {
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