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"## anonymous 5 years ago the limit as n -> infinity sqrt(n)[(sqrt(9n+1)-sqrt(9n)].
1. anonymous
looks as though that diverges unless there is supposed to be a division in that formula
2. anonymous
using the product rule f... | Let's delve into the concept of limits in calculus using the given text snippet as a starting point. We will break down the different aspects of this problem, including applying the product rule for limits and rationalizing expressions. Along the way, we will also discuss why some users suggested that the expression m... | ajibawa-2023/Maths-College |
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"# Differences
This shows you the differences between two versions of the page.
semirings_with_zero [2010/07/29 18:30]
127.0.0.1 external edit
semirings_with_zero [2019/03/28 16:22] (current)
jipsen
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-$\langle... | Semirings with zero are algebraic structures that consist of a set equipped with two binary operations, addition and multiplication, where the additive identity element, denoted by 0, behaves as a "zero." This means that adding 0 to any element does not change it, i.e., $a + 0 = a$, and multiplying any element by 0 re... | ajibawa-2023/Maths-College |
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"Question
# A man borrowed Rs.20000 at the rate of 10% per annum compounded annually. The total amount that needs to be paid at the end of one, two, three and four years is a, b, c and d respectively. An equation is defined s... | This problem involves the concept of compound interest, which is a type of interest that is calculated on both the initial principal and any accumulated interest from previous periods. It's important to note that the rate of interest remains constant over time, but the amount of interest earned increases as the balanc... | ajibawa-2023/Maths-College |
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"# Math Help - Integration: General Power Formula
1. ## Integration: General Power Formula
Hi. Question - integrate the following function:
28-1-19
(int) (4 + ((e^x))^3) e^x dx
The answer is 1/4(4 + (e^x))^4 + C
doing th... | In this explanation, we will derive the general power rule for integration and demonstrate its application through an example problem. The general power rule states that if you have a function of the form x^n where n is any real number, then its antiderivative is F(x)=(1/(n+1))x^(n+1)+C, where C is the constant of int... | ajibawa-2023/Maths-College |
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"# Parameterizing the upper hemisphere of a sphere with an upward pointing normal
Can someone explain how to do this?
area we're dealing with:
$x^2 + y^2 + z^2 = a^2, z \geq 0$
I'm aware that the answer is:
$x = a \sin(\p... | To understand where the parameterization of the upper hemisphere of a sphere comes from, let us first review the concept of spherical coordinates. Spherical coordinates are a three-dimensional coordinate system which defines any point in space based on its distance from the origin (radial distance), polar angle (θ), a... | ajibawa-2023/Maths-College |
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"Why Slopes of Perpendicular Lines Are Negative Reciprocals. Perpendicular lines are two lines that intersect and create 90 degree angles. thus, -7/3 Write an equation of the line that is perpendicular to 3x + 9y = 7 and passe... | Let's begin by discussing perpendicular lines and negative reciprocals, focusing on why slopes of perpendicular lines are negative reciprocals. Two non-vertical lines in a plane are considered perpendicular if they intersect at right angles (forming 90-degree angles). The slope of a line can be calculated using the fo... | ajibawa-2023/Maths-College |
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"# Category: Chapter 8
## Chapter 8 Exercise D
Exercise 1 By Exercise 11 in section 8B the characteristic polynomial is $z^4$ and by 8.46 this is a polynomial multiple of the minimal polynomial. Since $N^3 \neq 0$, it follow... | We begin our discussion with some fundamental concepts from linear algebra, specifically pertaining to chapter 8 of your course material. The objective here is to provide you with a clearer picture of two important ideas - the characteristic polynomial and the minimal polynomial of a linear operator. Moreover, we will... | ajibawa-2023/Maths-College |
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"# pdf
Probability density function for Gaussian mixture distribution
## Syntax
``y = pdf(gm,X)``
## Description
example
````y = pdf(gm,X)` returns the probability density function (pdf) of the Gaussian mixture distribut... | Probability Density Function (PDF) for Gaussian Mixture Distribution
-------------------------------------------------------------------
In statistics and machine learning, a Gaussian mixture model is a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian di... | ajibawa-2023/Maths-College |
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"# Discussion: Differing definitions for the rank of a set
I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.).
1. $\te... | The concept of the "rank of a set" is a fundamental notion in set theory, which deals with the hierarchical structure of sets. However, it appears that there are differing definitions for this term that can lead to confusion among learners. This discussion aims to clarify these distinctions by analyzing two prevalent ... | ajibawa-2023/Maths-College |
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"Or the line could completely lie inside the plane. To embed this widget in a post, install the Wolfram|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. Enter to functions. Point of In... | Let's delve into the fascinating world of geometry and linear algebra, specifically focusing on the concept of intersection of lines and planes. We will explore the mathematical underpings of these topics while providing clear and detailed explanations.
When discussing the intersection of geometric objects, it is cru... | ajibawa-2023/Maths-College |
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"# Mock AIME 1 2007-2008 Problems/Problem 13
## Problem
Let $F(x)$ be a polynomial such that $F(6) = 15$ and $$\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}$$ for $x \in \mathbb{R}$ such that both sides are defined. Find ... | This educational piece will delve into the problem-solving process of a challenging mathematics question, specifically focusing on polynomials. The ability to work through complex problems is essential for college students studying advanced math subjects like abstract algebra, calculus, or mathematical analysis. We'll... | ajibawa-2023/Maths-College |
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"## anonymous one year ago What is the simplified form of 4x^2-25/2x-5?
1. anonymous
@Michele_Laino
2. Michele_Laino
hint: we can factorize the numerator as follows: $\Large \frac{{4{x^2} - 25}}{{2x - 5}} = \frac{{\left( {... | The initial question posed asks for the simplified form of the rational expression (4x² - 25) / (2x - 5). To begin, let us review what it means to simplify a rational expression. Simplifying a rational expression entails factoring both the numerator and denominator completely and canceling out any common factors, if t... | ajibawa-2023/Maths-College |
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"# Eilenberg-Mac lane spaces and a generalization
Let $G$ and $H$ be two abelian groups and let $n>1, m>1$ be two different integers. How many different spaces $X$ (up to homotopy) do we have with the property $\pi_{n} X=G$ ,... | We begin our discussion with an examination of Eilenberg-Mac Lane spaces, also known as K(G,n), where G is an abelian group and n is a positive integer greater than 1. These spaces play a crucial role in algebraic topology because their only non-trivial homotopy group is the nth one, which is equal to G. This means th... | ajibawa-2023/Maths-College |
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"×
# You may've known about this before
$\frac{|x-y|}{\sqrt{x^2+1}\sqrt{y^2+1}}+\frac{|y-z|}{\sqrt{y^2+1}\sqrt{z^2+1}}\geq\frac{|z-x|}{\sqrt{z^2+1}\sqrt{x^2+1}}$
Let $$x,y,z\in\mathbb{R}$$, prove this inequality above.
Not... | The given inequality involves absolute values and expressions derived from the Cartesian coordinates of three points on a plane. To prove this inequality, let's first establish some definitions and properties associated with them. This will help build rigor into our discussion while ensuring a deep understanding of th... | ajibawa-2023/Maths-College |
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"• Magoosh
Study with Magoosh GMAT prep
Available with Beat the GMAT members only code
• FREE GMAT Exam
Know how you'd score today for $0 Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-acc... | When it comes to preparing for the Graduate Management Admission Test (GMAT), there are numerous resources available to help you study effectively and efficiently. Two highly regarded options are Magoosh and The Princeton Review (TPR). Both offer various tools and features designed to enhance your test preparation exp... | ajibawa-2023/Maths-College |
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"Homework Help: Laplace Transform of Step Function
1. Jun 21, 2012
ElijahRockers
1. The problem statement, all variables and given/known data
Solve
y'' + y = f(t), y(0)=0, y'(0)=1,
f(t)=
(0 for 0<t<pi)
(1 for pi<t<2pi)
(... | The Laplace transform is a powerful mathematical tool used to solve differential equations by converting them into algebraic equations. This method is particularly useful for solving problems like the one presented here, which involves a step function. However, as you've discovered, there are some subtleties involved ... | ajibawa-2023/Maths-College |
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"# Prove that $\lim_{n \to \infty} n^k(3 - x_n) = 0$, for any $k \in \mathbb{N}^*$
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence such that $x_{n+1} = \sqrt{3+2x_n}$ and $x_0 \in [0, 3]$. Prove that: $$\lim_{n \to \infty} n^k(3... | Now, let us delve into the problem at hand. We are given a sequence ${x_n}_{n\in \mathbb{N}}$, defined by the recurrence relation $x_{n+1} = \sqrt{3+2x_n}$ and $x_0 \in [0, 3]$. The task is to prove that for any positive integer $k$, the limit $$\lim\_{n \to \infty} n^k(3 - x_n) = 0.$$
Before diving into the main pro... | ajibawa-2023/Maths-College |
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"mathimatikoi.orghttps://www.mathimatikoi.org/forum/ Does there exist expression?https://www.mathimatikoi.org/forum/viewtopic.php?f=11&t=1223 Page 1 of 1
Author: Riemann [ Sat Jul 22, 2017 9:33 am ] Post subject: Does there ... | The given text snippet is about a problem posted on a mathematics forum (<https://www.mathimatikoi.org/forum/>) regarding the existence of certain expressions involving set theory and geometry. Specifically, it asks whether there exists an expression for the real plane, denoted as $\mathbb{R}^2$, as a union of infinit... | ajibawa-2023/Maths-College |
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"Homework Help: Probability Theory- Standard Normal Distributions
1. Sep 29, 2011
dzza
1. The problem statement, all variables and given/known data
Two types of coins are produced at a factory: a fair coin and a biased one... | To tackle this problem, we need to understand a few key concepts from probability theory, particularly those concerning the binomial distribution and its relationship to the standard normal distribution via the De Moivre-Laplace limit theorem. Let's explore these topics step by step.
First, let's recall the definitio... | ajibawa-2023/Maths-College |
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"# Math Help - Mathematical Induction
1. ## Mathematical Induction
Question:
Use the principal of Mathematical Induction to prove 2|(n2+n) for all n>=0
2. Originally Posted by oldguy
Question:
Use the principal of Mathema... | Mathematical induction is a fundamental proof technique in mathematics used to establish statements or formulas that hold for all natural numbers. It involves two steps: the base case and the inductive step. The base case demonstrates the statement's validity for the smallest possible value (often n = 0 or n = 1), whi... | ajibawa-2023/Maths-College |
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"# Why is base-10 the most common positional notation radix for number representation?
I understand that number can have multiple representations, and I can conceive that the positional notation system was better adapted for ... | The choice of base-10, or decimal system, as the most common positional notation radix for number representation has been attributed to the number of digits on humans' hands. While this explanation seems plausible and intuitive, demonstrating its validity requires historical evidence, ethnographic studies, or anthropo... | ajibawa-2023/Maths-College |
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"# What is wrong with my understanding about total pressure for a rocket nozzle?
I’m using the ideal rocket equations and isentropic flow but cannot determine what is wrong in my understanding about total pressure. Can anyone... | The confusion you're experiencing regarding total pressure in a rocket nozzle stems from the application of isentropic relations and the ideal rocket equation. Let's break down these concepts and examine your calculations step by step. This will help clarify any misconceptions and deepen your understanding of the prin... | ajibawa-2023/Maths-College |
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"## Clusterig overview
The goal in clustering is to take unlabeled data $$(x_i)_{i=1}^n$$, and produce a partition so that similar data is put in the same group, and dissimilar data is put in different groups.
Remark. Before... | Machine learning is often associated with supervised learning tasks such as classification and regression, where labeled data is used to train models to predict categorical or continuous outcomes, respectively. However, this view is limited and fails to capture the full scope of what machine learning has to offer. Uns... | ajibawa-2023/Maths-College |
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"# Rao-Blackwell exponential distribution
Let $X_1,..,X_n$ be a random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$
i) Find a unbiased estimator of $\lamb... | The Rao-Blackwell Theorem is a fundamental concept in statistical estimation theory. It provides a method to improve upon existing estimators by using sufficient statistics. Here, we'll address the student's concerns step-by-step while delving deeper into these concepts.
1. Unbiased Estimator (Part i): You correctly ... | ajibawa-2023/Maths-College |
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"# Complex Analysis and Limit point help
So S is a complex sequence (an from n=1 to infinity) has limit points which form a set E of limit points. How do I prove that every limit point of E are also members of the set E. I th... | To begin, let us establish a working definition of the terms involved in the problem:
* A \textit{complex sequence} is a function $f:\mathbb{N}\rightarrow\mathbb{C}$ where $\mathbb{N}=\left\{1,2,3,\ldots\right\}$ is the set of natural numbers and $\mathbb{C}$ denotes the set of complex numbers. We often denote the im... | ajibawa-2023/Maths-College |
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"Blog Entries: 5
## Why do people say that 1 and 0.999... are equal? Aren't they two different numbers?
Why do people say 1 and 0.999... are equal? Aren't they two different numbers?
No, they really are the same number, tho... | The equation 1 = 0.999..., where the right side goes on forever with repeating nines, is a common point of confusion for many students studying mathematics. This equality may seem counterintuitive because it visually appears that there is a gap between 0.999... and 1, leading some to believe that these are distinct nu... | ajibawa-2023/Maths-College |
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"# I need help with a varying refractive index problem
the refractive index varies as
$\mu = k*r$
find the least distance between light ray and centre of sphere
is this approach correct?
integration of dt i.e,ds/v is mini... | The problem at hand involves finding the least distance between a light ray and the center of a sphere, given that the refractive index μ varies as a function of the radial distance r from the center of the sphere. The refractive index is defined as the ratio of the speed of light in vacuum (c) to the speed of light i... | ajibawa-2023/Maths-College |
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"Share
Books Shortlist
# Solution for In Below Figure, Ab || Cd. If Oa = 3x – 19, Ob = X – 4, Oc = X – 3 and Od = 4, Find X. - CBSE Class 10 - Mathematics
#### Question
In below figure, AB || CD. If OA = 3x – 19, OB = x – ... | This problem involves finding the value of x using the properties of proportionality of diagonals in a trapezium. A trapezium is a quadrilateral with exactly one pair of parallel sides. The diagonals of a trapezium always intersect, dividing each other into ratios that are equal.
The given information includes line s... | ajibawa-2023/Maths-College |
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"# Solving the diffusion equation with state-dependent diffusion coefficient numerically?
I am having an issue numerically solving the following diffusion equation with state-dependent diffusion coefficient and need some help... | The diffusion equation is a partial differential equation (PDE) that describes how particles spread out from a concentrated area to areas of lower concentration over time. It has many applications in various fields such as physics, engineering, and finance. When the diffusion coefficient, which measures the rate of di... | ajibawa-2023/Maths-College |
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"# probability that the maximum and median occurs exactly once in set
#### MaryStew
##### New member
I was wondering if someone could confirm my answer for the 1st question and then maybe guide me how to do the 2nd question?... | To begin, let us review some fundamental concepts necessary for solving the given problems. A sequence of numbers is called uniformly distributed when every number in the set has an equal chance of being selected. Furthermore, the median of a sequence is the middle element when arranged in ascending order (if the leng... | ajibawa-2023/Maths-College |
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"# Velocity and Center of mass question?
1. Nov 14, 2011
### nukeman
Velocity and Center of mass question???
1. The problem statement, all variables and given/known data
Having some issues with this one.
"Ball A has a ma... | When dealing with collisions in physics, they can be classified as either elastic or inelastic. An elastic collision is one where there is no loss of kinetic energy, meaning that the total kinetic energy of the objects before the collision is equal to the total kinetic energy after the collision. On the other hand, an... | ajibawa-2023/Maths-College |
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"# Rotational motion problem
1. Nov 5, 2012
### vysero
1. The problem statement, all variables and given/known data
A rolling wheel moving to the right initially has an angular speed of 7.0 rad/s but after making 3
complete... | In physics, rotational motion refers to the movement of an object around a fixed axis. It involves several key parameters, including angular position, angular velocity, and angular acceleration. Understanding these concepts requires knowledge of both linear and rotational kinematics. This discussion will delve into th... | ajibawa-2023/Maths-College |
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"# A question about Choice Functions.
Assume the axioms of ZFC. Suppose that X is an infinite set of infinite (and pairwise disjoint) sets, none of which has a cardinal number greater than that of X. Is the cardinal number of... | In order to understand the statement provided, we first need to clarify some fundamental concepts in set theory, specifically those related to Zermelo-Fraenkel Set Theory (ZFC), cardinality, and choice functions. Afterward, we will analyze the given claim and its implications.
**1. Zermelo-Fraenkel Set Theory with th... | ajibawa-2023/Maths-College |
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"# Fungrim entry: 081188
${z + 1 \choose k + 1} = {z \choose k} + {z \choose k + 1}$
Assumptions:$z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, k \in \{0, 1, \ldots n\}$
TeX:
{z + 1 \choose k + 1} = {z \choose k} + {z \ch... | The equation presented is a property of binomial coefficients, which are also known as combinations. A binomial coefficient, denoted as ${n \choose k}$, represents the number of ways to choose $k$ items from a set containing $n$ distinct items without regard to order. This concept arises frequently in combinatorics an... | ajibawa-2023/Maths-College |
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"# Thread: If x^(2)=1 then G is abelian
1. ## If x^(2)=1 then G is abelian
I'm trying to show that if x^(2)=1 for all x in G then G is an abelian group.
Attempt at the solution:
xx=1
x=x^(-1)
xy=(xy)^(-1)
xy=y^(-1)x^(-1... | The thread you provided discusses a proof attempting to establish that if the square of every element in a group G equals the identity element 1, then G is an Abelian group. An Abelian group is a special type of group where the order of multiplication does not affect the result, meaning that the elements of the group ... | ajibawa-2023/Maths-College |
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"## Rectangular GLT sequences
Giovanni Barbarino, Carlo Garoni, Mariarosa Mazza, and Stefano Serra-Capizzano
### Abstract
The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the ... | Generalized Locally Toeplitz (GLT) sequences are a fundamental tool in the analysis of the asymptotic spectral distribution of square matrices that arise from the discretization of differential problems. As the mesh fineness parameter n increases to infinity, these sequences often converge to GLT sequences. This paper... | ajibawa-2023/Maths-College |
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"# Second order differential equation with function of $x$ as coefficients
Suppose that the second order linear equation
$$x^2y''+(3x+2x^3)y'-4y=0$$
has a fundamental set of solutions $\{y_1,y_2\}$ such that $y_1(1)=−3$, $y... | A second order differential equation (DE) has the general form $$a(x) y '' + b(x) y ' + c(x) y = f(x),$$ where $a(x)$, $b(x)$, $c(x)$, and $f(x)$ are functions of the independent variable $x$. When $f(x) = 0$, the DE is said to be homogeneous, otherwise it is nonhomogeneous. The given example is a homogeneous second o... | ajibawa-2023/Maths-College |
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"Saturday, October 11, 2014
A Strange Recursive Relation, Automatic
Hofstadter mentions the following recursive relation in his great book "Gödel, Escher, Bach": \begin{align} g(0) &= 0;\\ g(n) &= n-g(g(n-1)). \end{align} I ... | The passage provided introduces a recursive function named $g(n)$ and claims that it can be expressed as $\left\lfloor \phi\cdot (n+1) \right\rfloor$, where $\phi = \frac{-1+\sqrt{5}}{2}$. This section will delve into the details of the given information, focusing on rigorous explanations while maintaining engagement ... | ajibawa-2023/Maths-College |
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