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If $f(c)\neq 0$, then there exists $\delta\gt 0$ such that for all $x$, $|x-c|\lt\delta$, $|f(x)-f(c)|\lt \frac{|f(c)|}{2}$. In particular, $f(x)$ and $f(c)$ have the same sign.
But: if $f(c)\gt 0$, then this means that for all $\delta\gt 0$ there exists $s\in S$ such that $c-\delta \lt s\leq c$, so $f(s)\lt 0$, hence... | {
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So $z$ is in $Y$ and is a lower bound for $Y$, hence $z$ is the least element of $Y$. Thus, $z$ is the least upper bound of $S$, and $S$ has a least upper bound, as desired. QED
The importance is not so much because of the fact that the least upper bound property itself can be used directly to prove major thoerems. As... | {
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One could take other axioms as a starting point for analysis. For example, completeness for metric spaces is formulated in terms of Cauchy sequences, mentioned in Arturo's and Thomas's answers, and that can also be done for $\mathbb R$. There is one difference, however, which is that convergence of Cauchy sequences is ... | {
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The fact that every bounded non-decreasing sequence has a limit is a direct result of this theorem.
The fact that every bounded sequence has a convergent subsequence is a relatively direct consequence.
From there, you can get that any Cauchy sequence is convergent.
There are two essential ways to define the reals fr... | {
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# Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?
Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?
I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= \{a+bi \mid a,b \in \mathbb{Q}\} \end{align*}
I tend to bel... | {
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Alternatively, show that $$\Bbb Q\subseteq\Bbb Q(i)\cap\Bbb Q\bigl(\sqrt2\bigr),$$ which I leave to you. Then, suppose $z\in\Bbb Q(i)\cap\Bbb Q\bigl(\sqrt2\bigr).$ Since $z\in\Bbb Q\bigl(\sqrt2\bigr),$ then $z\in\Bbb R.$ From there, we can use the fact that $z\in\Bbb Q(i)$ to readily show that $z\in\Bbb Q,$ completing ... | {
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# The Stacks Project
## Tag: 01CV
This tag has label modules-definition-gamma-star and it points to
The corresponding content:
Definition 16.21.4. Let $(X, \mathcal{O}_X)$ be a ringed space. Assume that all stalks $\mathcal{O}_{X, x}$ are local rings. Given an invertible sheaf $\mathcal{L}$ on $X$ we define the ass... | {
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Comment #13 by Pieter Belmans on July 22, 2012 at 9:23 am UTC
Missed this earlier on: there is the typo "compatibilties" in the last sentence of the remarks following this definition.
Comment #20 by Johan on July 22, 2012 at 10:58 pm UTC
@#11 and #13. Fixed. Thanks. @#12: Yes, hmm, I think of the notation $\Gamma_*$... | {
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# Seeking Methods to solve $I = \int_{0}^{\frac{\pi}{2}} \frac{\arctan\left(\sin(x)\right)}{\sin(x)}\:dx$
I was wondering what methods people knew of to solve the following definite integral? I have found a method using Feynman's Trick (see below) but am curious as to whether there are other Feynman's Tricks and/or Me... | {
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\begin{align} \int_0^{\pi/2}\frac{\arctan \sin(x)}{\sin(x)}dx &=\int_0^{\pi/2}\frac{1}{\sin(x)}\sum_{n=0}^\infty \frac{(-1)^n \sin^{2n+1}(x)}{2n+1}dx\\ &=\sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \int_0^{\pi/2}\sin^{2n}(x)dx\\ &=\frac{\pi}{2}+\frac{\pi}{2}\sum_{n=1}^\infty \frac{(-1)^n}{2n+1}\cdot \frac{(2n-1)!!}{(2n)!!}\\... | {
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$$I = \int_{0}^{1}\frac{\arctan x}{x\sqrt{1-x^2}}\,dx =\sum_{n\geq 0}\frac{(-1)^n}{2n+1}\int_{0}^{1}\frac{x^{2n}}{\sqrt{1-x^2}}\,dx=\frac{\pi}{2}\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)}\cdot\frac{\binom{2n}{n}}{4^n}$$ is a fairly simple hypergeometric series, namely $$\frac{\pi}{2}\cdot\phantom{}_2 F_1\left(\tfrac{1}{2},\tf... | {
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Slightly different from @Frpzzd's answer $$I=\int_0^{\pi/2}\frac{\arctan\sin x}{\sin x}\mathrm dx$$ Recall that $$\arctan x=\sum_{n\geq0}(-1)^n\frac{x^{2n+1}}{2n+1},\qquad |x|\leq1$$ And since $$\forall x\in\Bbb R ,\ \ |\sin x|\leq1$$, we have that $$\arctan\sin x=\sum_{n\geq0}\frac{(-1)^n}{2n+1}\sin(x)^{2n+1},\qquad \... | {
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It is easily shown that $$\mathrm{D}^nx^\alpha=p(\alpha,n)x^{\alpha-n}$$ Where $$p(\alpha,n)=\prod_{k=1}^{n}(\alpha-k+1)$$ is the falling factorial. Hence $$\mathrm{D}_{x=1}^nx^\alpha=p(\alpha,n)$$ So $$x^{\alpha}=\sum_{n\geq0}\frac{p(\alpha,n)}{n!}(x-1)^n$$ $$(1+x)^{\alpha}=\sum_{n\geq0}\frac{p(\alpha,n)}{n!}x^n$$ The... | {
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# How to add and subtract values from an average?
Say I have 100 numbers that are averaged:
number of values = 100
total sum of values = 2000
mean = 2000 / 100 => 20
If I want to add a value and find out the new average:
total sum of values = 2000 + 100
mean = 2100 / 101 => 20.79
If I want to subtract a value an... | {
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• And $ns$ is the old sum, as you can derive from the first equation. Feb 16, 2011 at 13:21
• I'm tempted to downvote as ns is not explained in the answer Aug 22, 2014 at 20:32
• At first I commented that @501's answer was clearer but realized I didn't fully understand why the question worked. This answer, while denser... | {
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Is it possible to sort an array of 10 elements with 20 comparisons of two elements?
I think it's not, because there are $10!$ permutations of $10$ elements, and $10! \gt 2^{20}$.
Can anyone point me how to prove it rigorously?
-
It depends on the sorting algorithm used. – npisinp May 19 '14 at 16:50
The best compari... | {
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For a list with 10 elements there are $10! = 3628800$ possible input lists and therefore, for any given comparison sort, two such lists must arrive at the same leaf, and the sort therefore uses the same sequence of swaps to order the items in both lists. But this must leave at least one of them out of order.
This all ... | {
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Geometric sequence examples
### Geometric sequence examples | {
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Seq. Geometric sequences are used in several branches of applied mathematics to engineering, sciences, computer sciences, biology, finance Problems and exercises involving geometric sequences, along with detailed solutions and answers, are presented. The formula is then used to find another term of the sequence. For ex... | {
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fraction: T he sum of an infinite geometric sequence, infinite geometric series This unit introduces sequences and series, and gives some simple examples of each. 01) Month 5 Research Sources a = (100) (1. Geometric Sequence Calculator Find indices, sums and common ratio of a geometric sequence step-by-step Geometric s... | {
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A sequence is a list of numbers or objects, called terms, in a certain order. Here the ratio of any two terms is 1/2 , and the series terms values get increased by factor of 1/2. In variables, it looks like In variables, it looks like Geometric Examples. Example 1. Now let’s have a look at some examples where we can us... | {
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difference, or a constant difference between each term. 5 . Geometric series. 625 + 0. A sequence made by multiplying by the same value each time. It's because it is a different kind of a sequence - a geometric progression . An infinite geometric sequence is a geometric sequence Problems. Then give a recursive definiti... | {
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the equation is evaluating. An arithmetic sequence is a sequence with the difference between two consecutive terms constant. Be careful here. A geometric sequence (geometric progression) is defined as a sequence in which the quotient of any two consecutive terms is a constant. Since a geometric sequence is a sequence, ... | {
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can be anything - but typically arithmetic. (a) Show that every term of a geometric sequence with non-negative terms, except the first term and the last term (in case of a finite sequence), is the geometric average of the preceding term and the following term. ARITHMETIC AND GEOMETRIC SEQUENCE WORD PROBLEMS PRACTICE. com... | {
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after the last one given. We dealt a little bit with geometric series in the last section; Example 1 …Illustrated definition of Geometric Sequence: A sequence made by multiplying by the same value each time. So once you know the common ratio in a geometric sequence you can write the recursive form for that sequence. 8+... | {
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where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Hence the geometric sequence will be . Using Explicit Formulas for Geometric Sequences. This video shows how derive the formula to find the 'n-th' term of a geometric sequence by considering an ... | {
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the (k+i)th term for all i such that these terms exist. Since n ! > n, our intuition suggests that the sequence { n !} n=1,2,3, Exponential. Step (2) The given series starts the summation at , so we shift the index of summation by one: Our sum is now in the form of a geometric series with a = 1, r = -2/3. Learn more ab... | {
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ric sequences. In mathematics, a sequence is usually meant to be a progression of numbers with a clear starting point. LIMITS OF RECURSIVE SEQUENCES 3 Two simple examples of recursive definitions are for arithmetic sequences and geomet- ric sequences. The sequence <1,2,4,8,16,… = is a geometric sequence with common rati... | {
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sequence is a list of numbers in which each number depends on the one before it. The equality given in Example 4. Homework problems on geometric sequences often ask us to find the nth term of a sequence using a formula. Given the first term and the common ratio of a geometric sequence find the first five terms and the ... | {
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in Exs. If we multiply, it is a geometric sequence. Ask students to find the patterns. a sequence (such as 1, 1/2, 1/4) in which the ratio of a term to its predecessor is always the same —called also geometrical progression,…A geometric sequence is an ordered list of numbers in which each term is the product of the pre... | {
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rate is calculated as ((1+0. 1, 0, 3, 0, 5, 0, 7, Arithmetic and Geometric Sequences. 2: Compound Interest; Geometric Sequences De nitions: If $P is invested at an interest rate of r per year, compounded annually, the future A sequence is a list of numbers, geometric shapes or other objects, that follow a specific patt... | {
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1, 3, 5, 7, d=2 10, 20, 30, d=10 10, 5, 0, 5,. This ratio is called the common ratio. Instructor: Dr. Here are a few examples of geometric sequences. Example 4. Algebra > Sequences and Series > Geometric Sequences. An arithmetic sequence is a list of numbers with a definite pattern. Each term (except the first term) i... | {
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for a sequence tells you the value of the n th term as a function of . In a Geometric Sequence each term is found by multiplying the previous term by a constant. Full Answer. The sides of the cuboid make a geometric progression. The value of the $$n^{th}$$ term of the arithmetic sequence, $$a_n$$ is computed by using t... | {
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for an arithmetic sequence is the value being added between terms, and is represented by the variable d. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2. Concept 16 Arithmetic & Geometric Se... | {
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whose rule is the multiplication of a constant are called geometric sequences, similar to arithmetic sequences that follow a rule of addition. Primary SOL . 70 and 71. c) Find the value of the 15 th term. Are the following sequences arithmetic, geometric, or neither? If they are arithmetic, state the value of d. Theref... | {
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it an arithmetic sequence. 80, and x = 4: P(X=4)=0. That's our total number of terms. 12, which is known as the ratio test. The sequence <1,2,4,8,16,… = is a geometric sequence with common ratio 2, since each term is obtained from the preceding one by doubling. 444 5. Geometric Series. Finite geometric series applicati... | {
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starting number 3, common factor 2 and eight terms is 3, 6, 12, 24, 48, 96, 192, 384. Here are the all important examples on Geometric Series. The geometric sequence can be rewritten as where is the amount Infinite Geometric Sequences. OK, so I have to admit that this is sort of a play on words since each element in a ... | {
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in the sequence a term. Explain that patterns (sequences) can occur with a list of numbers. 2)*(1+0. for some q> 0. an Dan1 Cd or an an1 Dd: The common difference, d, is analogous to the slope of a line. In an . If you're going on to Calculus, these are going to be important! Remember that with arithmetic sequences we ... | {
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Sometimes the terms of a geometric sequence get so large that you may need to express the terms in scientific notation rounded to the nearest tenth. Geometric sequence example. There are exceptions of course like the ball bouncing is geometric even though it is singular because of coefficient of restitution. So for exa... | {
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sequence example convergent geometric series the sum of an infinite geometric recursive formulas for geometric sequences practice khan academy solved given the first three terms of a geometric sequence fi geometric sequence formula examples video lesson transcript ppt section 57 arithmetic and geometric sequences power... | {
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common ratio in each of the following geometric sequences. An example would be 3, 6, 12, 24, 48, … Each term is equal to the prior one multiplied by 2. Thus making both of these sequences easy to use, and allowing us to generate a formula that will enable us to find the sum in just a few simple steps. The more general ... | {
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term IntroExamples Arith. com. 1 Sequences and Series. 3. Another way to look at it is that we are multiplying each term by ½ to get the next term in the sequence. Examples of arithmetic and geometric sequences and series in daily life. 1)-4, 1, 6, 11, … 2) 2 Solutions for the assessment 7. Geometric Series Examples Ge... | {
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in Example 2) we find that the general term for the denominator is a n 974 Chapter 10 Sequences, Induction, and Probability Thus, the formula for the term is a n =a 1 r n–1. A recursive formula allows us to find any term of a geometric sequence by using the previous term. The formula is broken down into a 1 which is th... | {
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progression) • To enable students apply their knowledge of geometric sequences to everyday applications If»students»are»unable»to»suggest»examples»of» geometric»sequences»direct»them»to»examples» 4. 4. notebook April 25, 2014 IF Checking: p. In this case, 2 is called the common ratio of the sequence. A geometric sequen... | {
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each term is obtained by adding a fixed number to the previous term. A sequence is a set of numbers determined as either arithmetic, geometric, or neither. Examples of How to Apply the Concept of Arithmetic Sequence. So this sequence, which is not a geometric sequence, we can still define it explicitly. The student pop... | {
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Geometric Sequences Worksheet Determine whether each of the following sequences is arithmetic, geometric, or neither. Sequences and series, whether they be arithmetic or geometric, have may applications to situations you may not think of as being related to sequences or series. Find the sum of the first six terms of th... | {
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a sequence like . For example, in the sequence 2/1, 4/2, 6/3, the common ration is 2. Plugging those values into the general form of the geometric sequence (as done in Example 2) we find that the general term for the denominator is a n = 2 (2) n-1 = 2 n. The individual items in the sequence are called terms , and repre... | {
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a formula for the nth term of the geometric sequence 6, 2, 2 3, 2 9, . Examples: d = the common difference . As a function of q, this is the Riemann zeta function ζ(q). phpExample problems that can be solved with this calculator. The lesson includes three examples for the teacher to use and a two page worksheet for stu... | {
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with this calculator. The geometric sequence is sometimes called the geometric progression or GP, for short. Each number of the sequence is given by multipling the previous one for the common ratio. This algebra lesson explains geometric sequences. It's not a geometric sequence, but it is a sequence. In this lesson, it... | {
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examples video lesson transcript ppt section 57 arithmetic and geometric sequences powerpoint 9 Introduction. a = (a ) (1. The geometric mean is similar to the arithmetic mean. Well, our website offers hundreds of free examples of geometric sequences. Example 4: Full Answer. Given the first term and the common ratio of... | {
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formula is given by Here a will be the first term and r is the common ratio for all the terms, n is number of terms. Apr 08, 2010 · An introduction to geometric sequences Practice this lesson yourself on KhanAcademy. is a geometric sequence with r = 3. org right now: https://www. 25 + 0. Example 3: Find the next three ... | {
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the first four terms and the 10th term of a geometric sequence with a first The recursive formula for a geometric sequence is written in the form For example, when writing the general explicit formula, n is the variable and does not Example: 1, 2, 4, 8, 16, 32, 64, 128, 256, This sequence has a factor of 2 between each... | {
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with arithmetic and geometric sequences and series, but give them neither explanations of when to use the formulas nor the meanings of the symbols. Arithmetic sequences have this same special property: equal changes in the input (e. Geometric Sequence. For convenience, at times we use …Example (continued) Solution. com... | {
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3 for which we try to figure out which is the 10 th number in the sequence: ■ The 10 th value of the sequence (a 10) is 39,366 ■ Sample of the first ten numbers in the geometric sequence: 2; 6; 18; 54; 162; 486; 1,458; 4,374; 13,122; 39,366. An arithmetic sequence (arithmetic progression) is defined as a sequence of nu... | {
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# Solve the system of inequalities
Solve the system: $$|3x+2|\geq4|x-1|$$
$$\frac{x^{2}+x-2}{2+3x-2x^{2}}\leq 0$$
So for $|3x+2|\geq4|x-1|$ I got the solution $x=6$ and $x=2/7$ and Wolframalpha agrees with me. I'm having troubles writing the final solution for $\frac{x^{2}+x-2}{2+3x-2x^{2}}\leq 0$.
$1)$ $x^{2}+x-2\... | {
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(The idea is to solve $x^2+x-2=0$ and $2+3x-2x^2=0$ and study the inequality on each region you obtain. Note that we can't divide by $0.$ So $x\ne-1/2$ and $x\ne 2.$)
Finally, one gets the intersection to obtain $(2/7,1]\cup (2,6].$
• Thanks, I appreciate your help! – lmc Oct 23 '16 at 13:08
• @mfl: I think you have ... | {
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# Finding the probability that an ace is found in every pile when a deck of cards is split into 4
I'm trying to answer this question and you are supposed to use the multiplication rule to solve it:
A deck of 52 playing cards is randomly divided into four piles of 13 cards each. Compute the probability that each pile ... | {
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so by the multiplication rule I get that $$\mathbb{P}(A_{1} \cap A_{2} \cap A_{3} \cap A_{4}) = \frac{1406}{4165} \frac{225}{703} \frac{13}{50} \approx 0.0281$$
However the answer I am given says it should be $\approx 0.105$. Can anyone help me to see where I have gone wrong? Would it perhaps be that defining the even... | {
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As desired. So, you did the basic method largely correct, but you did some sloppy calculations, and more importantly, you forgot to take into account that for the first few piles you have a choice of aces.
ETA: I see that the use of the multiplication rule is expected. I'm still going to leave this here as an approach... | {
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The $1^{st}$ ace can go to any slot, and given that, the $2^{nd}$ ace has $39$ out of $51$ free slots to go in,
and so on and so forth, thus simply$\quad\frac{39}{51}\cdot\frac{26}{50}\cdot\frac{13}{49}$
This way, we need not bother at all as to how the other $48$ cards are distributed.
A different approach:
There a... | {
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# Calculating percentage that's equal to a value
Let's say I have a normal distribution with people lengths in which the mean is 170 cm and the standard deviation is 6 cm.
I know you can calculate the lower tail and higher tail of 180 cm with
pnorm(180, mean=170.6, sd=6.75, lower.tail=TRUE/FALSE)
But how would I g... | {
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The R code:
library(ggplot2)
ggplot(data.frame(x = c(150, 190)), aes(x = x)) +
stat_function(xlim = c(150, 180), fun = dnorm, args = list(mean = 170.6, sd = 6.75),
geom = "area", alpha = 0.5, fill = "red") +
stat_function(xlim = c(150, 170), fun = dnorm, args = list(mean = 170.6, sd = 6.75),
geom = "area", alpha = 0.5... | {
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The obtained probability is 0. This happens because the normal distribution is a continuous probability distribution.
Don't hesitate to ask if you have further doubts. | {
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# An interesting integral $I = \int\limits_{-1}^{1} \arctan(e^x)dx$
I solved this interesting integral online: $$I = \int\limits_{-1}^{1} \arctan(e^x)dx$$ Now I tried the substitution $u=e^x$ but it lead me nowhere. I was looking at the following post which was solved in a beautiful way Integrate $\int_0^{\pi/2} \frac... | {
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Also, I think we can extend this to a broader result where we replace $x$ by any arbitrary odd function $g(x)$ and show that $$I = \int\limits_{-a}^{a} \arctan(e^{g(x)})dx = \frac{a \pi}{2}$$ for any odd function $f(x): (-a,a) \to\Bbb R$. Essentially the proof for this would follow the exact same reason as above right?... | {
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I have found another way to integrate this integral,first you have to know this identity, very useful for integrals with arctan $$\arctan(y)+arctan(1/y) =\frac{\pi}{2}\qquad (1)$$ with x positive or equal to 0, and $$\arctan(y)+arctan(1/y) =\frac{-\pi}{2}$$ for y negative. The first step is separate the integral
$$\in... | {
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# Strong Induction to find all possible combinations of two numbers
Full disclosure, this is a homework question, so I'm only looking for hints not full solutions please.
There is a store which offers two denominations of gift certificates, \$25 and \$40. Determine the possible total amounts that can be formed using ... | {
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Notice that $5$ and $8$ are coprime and thus their non-negative integer linear combinations form a numerical semigroup with two generators. A nice result of numerical semigroups with two generators $m,n$ is that any number bigger than $mn-m-n$ is in the semigroup.
To show this by induction for this particular problem,... | {
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Then you're left with the small cases, which you'll need to check by hand | {
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# Why does $a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converge to an irrational number?
There is a problem in my textbook that goes like this
$$a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$
and
$$a_0 =1$$
for all $n\ge1$.
It is monotonically decreasing sequence of rational numbers and bounded below. Howeve... | {
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$$a''=\frac{a+a'}2=\frac{a+\dfrac sa}{2}.$$
As can be shown, when you are close to the root, the sequence converges extremely rapidly.
For example,
$$a=\color{green}{1.41}\implies a'=\color{green}{1.41}84397163121\cdots\implies a''=\color{green}{1.41421}9858156\cdots$$ while the true value is $$\sqrt2=1.414213562373... | {
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• This is very nice but it doesn't seem obvious that the approximations converge. While the average is better than the worse of the two approximations it might be much worse than the better of the two approximations so you might actually be diverging if you choose the average as your new approximation. Or am I missing ... | {
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Suppose $f(x)=x^2-2$ so roots are $\pm \sqrt2$ now use Newton method $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ so you will have $$x_{n+1}=x_n-\frac{x_n^2-2}{2x_n}\\=\frac{\frac{2x_n^2-x_n^2+2}{x_n}}{2}\\=\frac{x_n+\frac{2}{x_n}}{2}$$ now take $x_n \to a_n$ so $$a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ and note $a_n... | {
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# Math Help - Balls in an Urn
1. ## Balls in an Urn
Twenty balls are placed in an urn. Five are red, five green, five yellow and five blue. Three balls are drawn from the urn at random without replacement. Write down expressions for
the probabilities of the following events. (You need not calculate their numerical va... | {
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There are: . ${20\choose3} \,=\,1140$ possible outcomes.
There are: 5 Reds and 15 Others.
We want 1 Red and 2 Others. .There are: . ${5\choose1}{15\choose2} \:=\:525$ ways.
. . Therefore: . $P(\text{exactly 1 Red}) \:=\:\frac{525}{1140} \:=\:\frac{35}{76}$
Select 3 of the 4 colors: . ${4\choose3} \:=\:4$ ways.
To p... | {
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Red-Green-Blue, Red-Green-Yellow, Red-Blue-Yellow, and Green-Yellow-Blue. Each of those three color combinations has 125 different arragements, because as we established above, there are 125 ways of arranging Color1-Color2-Color3. Therefore there are 500 total ways of arranging 3 out of 4 colors (where there are 5 uniq... | {
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. . Therefore: . $P(\text{Blue} > \text{Yellow}) \:=\:\frac{60}{1140} \;=\;\frac{1}{19}$
Not the ones I mentioned as well. When I had worked the answer out, I included these 2 cases, along with the cases I mentioned to get the answer.
I want to know why those are not needed.
8. I think I understand your question. Yo... | {
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# $\left(-\frac{1}{2}\right)! = \sqrt{\pi}?$ [duplicate]
I recently learned that $\left(-\frac{1}{2}\right)! = \sqrt{\pi}$ but I don't understand how that makes sense. Can someone please explain how this is possible?
Thanks!
-
Look up the gamma function on Wikipedia. – Ted Shifrin Jun 16 '13 at 20:31
That's because... | {
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What follows is the answer I posted there. Several others also posted good answers.
If there's any justice in the universe, someone must have asked here how to show that $$\int_{-\infty}^\infty e^{-x^2/2}\,dx = \sqrt{2\pi}.$$ Let's suppose that has been answered here. Let (capital) $X$ be a random variable whose proba... | {
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-
factorial of negative numbers are $defined$ in terms of Gamma function, i.e. $(x!) := \Gamma(x+1)\forall x\in\mathbb R$, except for $x$ to be negative integer. this is because the two of them agree on positive integers, and so this is just a convention.
Actually, $\Gamma(n+1)=n!$ for $n\in\mathbb{Z}$ and neither exis... | {
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# Math Help - exponential functions
1. ## exponential functions
1.Tiger drops a golf ball from a height of 60 m. After each bounce, the height of the balls bounce is 4/5 the height of the previous bounce. How high does the ball bounce after the tenth bounce? (Set up an exponential equation for the height.)
2.The dee... | {
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1. how do i set up the exponential equation for that?
2. no graphing software should be needed
4. Originally Posted by william
1. how do i set up the exponential equation for that?
That's the pattern you're looking for.
Originally Posted by william
2. no graphing software should be needed
Okay. Start by setting up v... | {
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The population $n$ years after 2005 is given by: . $P(n) \:=\:225\left(\frac{6}{5}\right)^{n-1}$
In 2015 $(n = 10)\!:\;\;P(10) \:=\:225\left(\frac{6}{5}\right)^9 \:\approx\;1161$ deer.
6. Originally Posted by Soroban
Hello, william;!
The height $h$ at the $n^{th}$ bounce is given by: . $h(n) \:=\:60\left(\frac{4}{5}... | {
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# Does $AB+BA=0$ imply $AB=BA=0$ when $A$ is real semidefinite matrix?
This is a general question that came to my mind while listening to a lecture(although its framing may make it look like a textbook question).
Suppose that $A$ and $B$ be real matrices. $A$ is symmetric and positive semi-definite$(x^tAx\ge0\ \ \for... | {
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• Either $v$ is an eigenvector with eigenvalue zero, in which case $A(Bv) = 0$, This means $ABv = BAv = 0$ for all eigenvectors corresponding to eigenvalue zero.
• Or $v$ is an eigenvector with eigenvalue $\lambda>0$. Suppose $Bv\neq 0$, then $Bv$, by above observation is an eigenvector of $A$ with negative eigenvalue!... | {
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An alternative proof is as follows. Since $A$ is psd, $A=UDU^*$, where $U$ is unitary and $D$ is diagonal. Suppose $A$ has $k$ distinct eigenvalues, $\lambda_1 < \lambda_2 <...< \lambda_k$. Then $D$ may be chosen to be $D=\lambda_1I_{m_1}\oplus\lambda_2I_{m_2}\oplus...\oplus\lambda_kI_{m_k}$, where $m_i$ denotes the mu... | {
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# Math Help - Party Question
1. ## Party Question
I did this by hand
for n =1, I get 1 way
n=2, 3 ways
n=3, 15 ways.
Any help or advice on how to look at this problem.
2. The number of ways to arrange 2n people into n named groups of two is $
\left( {\begin{array}{c} {2n} \\ {\underbrace {2,2, \cdots ,2}_n} \\ \... | {
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The numerator is simply 9! which is (2n-1)! and so I looked at the denominator for a while, and realized that if I divided those numbers by 2, I would have 4! Which means that if I multiplied 4! by $2^4$ I would have the denominator. Basically I made it look like this: $8*6*4*2 =4(2)*3(2)*2(2)*1(2) = (4*3*2*1)(2*2*2*2)... | {
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5. Originally Posted by Jrb599
Angel White... That was amazing help
I actually had the formula
1*3*5*---*n,
couldn't think of how to write it.
Thanks so much!
Btw, your explanation was awesome
If wanted to do this for 2n+1 people, and 1 person was left out, then I would get:
$
f(n) = \frac{(2n+1)!}{(n)!*2^{n}}
... | {
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6. Angel white, if you see my post early, that's what I got too so I would take it as right also. It took me a while to see why, but your post is much better at understanding the even way. Plato's posted helped me see it the odd way. The only reason why I asked for you to confirm is because I thought that you would off... | {
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# Find an infinite set $S$ and a function $g : S \to S$ that is surjective but not injective.
Find an infinite set $S$ and a function $g : S \to S$ that is surjective but not injective.
This is all that is given in the problem. Should I fix $S$ to be a certain set, like the integers, or natural numbers, and work from... | {
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Try to come out with a rule to describe what I did.
• Right. So 0-->0, 1-->0, 2-->1, 3-->1, 4-->2, 5-->2, etc... This would be an infinite set S mapped to itself, being surjective and not injective. Now to figure out what the rule is. – mathmajor Nov 27 '17 at 21:59
• Would this be a piecewise function? – mathmajor No... | {
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Then $$f(n +1) = n$$ for all $$n ∈ \mathbb{N}$$, so $$f$$ is surjective.
But $$f(1) = 1 = f(2)$$ and $$1 ≠ 2$$.
Therefore $$f$$ is not injective.
This website provide really good answer https://mathgraphy.com/functions/ | {
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# Prove: if $f(0)=0$ and $f'(0)=0$ then $f''(0)\geq 0$
let $$f$$ be a nonnegative and differentiable twice in the interval $$[-1,1]$$
Prove: if $$f(0)=0$$ and $$f'(0)=0$$ then $$f''(0)\geq 0$$
1. Are all the assumptions on $$f$$ necessary for the result to hold ?
2. what can be said if $$f''(0)= 0$$ ?
Looking at t... | {
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As can be seen from the arguments above we don't need continuity of $f''$. And we can avoid difficult theorems like Taylor or L'Hospital. The argument used in first paragraph leads to one of the simpler proofs of Taylor's Theorem with Peano's form of remainder.
Update: The condition $f' (0)=0$ is superfluous here. The... | {
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For 1) relaxing the condition that $f(0)=0$, we could look at $f(x)=\cos(x)+3$ (which has instead $f(0)=4$). It satisfies the property that $f(x)$ is non-negative, is twice differentiable on $[-1,1]$ and that $f'(0)=0$. However, $f''(0)=-\cos(0)=-1$ is not non-negative.
Relaxing the condition that $f(x)$ be non-negati... | {
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"lm_q2_score": 0.8791467595934565,
"openwebmath_perplexity": 212.45621806541004,
"openwebmath_score": 0.9331564903259277,
"tags... |
The continuity of $f''$ is not really needed to apply Taylor's formula with the Lagrange remainder. Anyway, we may just apply l'Hopital's rule to get: $$f''(0) =\lim_{x\to 0}\frac{f'(x)-f'(0)}{x} = 2\lim_{x\to 0}\frac{f(x)}{x^2}.$$ Since both $f(x)$ and $x^2$ are non-negative over $[-1,1]\setminus\{0\}$, $\;f''(0)\geq ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.96036116089903,
"lm_q1q2_score": 0.8442984026437923,
"lm_q2_score": 0.8791467595934565,
"openwebmath_perplexity": 212.45621806541004,
"openwebmath_score": 0.9331564903259277,
"tags... |
# Conditional Probability against Combinatorics
Consider a lot consisting of 3 blue balls and 1 red ball.
Suppose I pick 2 balls one after another without replacement, now the probability of 2 balls being blue:
$$\frac{3\choose 2}{4 \choose 2}$$
Now taking different approach using conditional probability, the solut... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145707166789,
"lm_q1q2_score": 0.8442976459012491,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 419.48645035666584,
"openwebmath_score": 0.8306035995483398,
"ta... |
Both approaches are correct. Both methods can be viewed as combinatorial.
In your first approach, your sample space consists of ${4 \choose 2} = 6$ unordered samples, each with probability $1/6.$ Three of the six outcomes correspond to choosing two blue balls, for a probability of $1/2.$ It is OK to use unordered samp... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145707166789,
"lm_q1q2_score": 0.8442976459012491,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 419.48645035666584,
"openwebmath_score": 0.8306035995483398,
"ta... |
Inference: There is hell a lot of way to get solution to a probability question,all the answers match provided the approach is right.Everyone chooses their own approach,so there is no "efficient" approach. This is also a reason why I(Most of us) love it. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145707166789,
"lm_q1q2_score": 0.8442976459012491,
"lm_q2_score": 0.8519528076067262,
"openwebmath_perplexity": 419.48645035666584,
"openwebmath_score": 0.8306035995483398,
"ta... |
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