Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

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Convergence in probability exercise 2

I have the following problem: Let $X_{n}$, $n \in \{1,2,...\}$. If the density of $X_{n}$ are $f_{n} = \left\{ \begin{array}{ll} \frac{n\cos(nx)}{\sin(n\theta)} & \mbox{if } 0 < x < \...
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Exercise about almost surely convergence

I have the following problem: Let $X_n$, $n \geq 1$ be independent r.v. identically distributed with the probability function: $P(X_{i} = k) = P(X_{i} = - k) = \frac{p}{2} q^{k-1}$, $k = 1,2,...,m$, ...
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Proving that a "$1 - (1 - (\frac1m))^n$" Converges to $0.6321205588...$ when $m = n =$ large

Suppose there a "$m$" objects One of these "$m$" objects is of interest (called "$m_0$") If you randomly select one of these "$m$" objects and then put it back, ...
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Method for convergence with unknown constant

I was expanding on specific telescopic series $$\sum_{n=1}^\infty \frac{1}{n(n+1)}= 1$$ $$\sum_{n=1}^\infty \frac{1}{n(n+2)}= \frac{3}{4}$$ $$\sum_{n=1}^\infty \frac{1}{n(n+3)}= \frac{11}{18}$$ It ...
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Smooth function $C^{\infty}$ that in the limit becomes Coulomb potential

I need to have a scalar function $\psi(\mathbf{r}, \sigma)$ of the coordinate $\mathbf{r}$ in euclidean 3-D space, which depends on a scalar adjustment parameter $\sigma$, such that in the limit $\...
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1answer
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Limit involving Stirling numbers of the first kind

This question arose whilst working on this problem. Let $\ \left[{n\atop k}\right]\ $ denote the $\ (n,k)-$th Stirling number of the first kind. Define $$ f(n) = \left[{n\atop 1}\right] r\ +\ \left[{n\...
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The sequence of distances of a sequence with no limit from a compact set converges to zero

Let $(X,d)$ is a metric space and $U \subset X$ is an open set such that $X- U$ is compact, where $X-U:=\{x \in X: x \notin U\}$. Let $A \subset U$ is a countable infinite subset of $U$ such that $A$ ...
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Find all function $f$ such that, for any sequence $(x_n)$ Cesàro-convergent, the sequence $(f (x_n))$ is also Cesàro-convergent

Let's say that a function $f$ is Cesàro-continuous at $x_0$ iff for any sequence $(u_n)_\Bbb{N}$ whose Cesàro mean converges to $x_0$, the Cesàro mean of the sequence $(f(u_n))_\Bbb{N}$ converges to $...
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$a_{n+1} = a_n/2 + 1/a_n$ is Cauchy but has no limit in $\mathbb{Q}$

I want to show that the sequence recursively defined by $a_{n+1} = \frac{a_n}{2} + \frac{1}{a_n}, \:\: a_1=1$ is a Cauchy sequence that does not converge in $\mathbb{Q}$. My idea was to show that ...
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Does expected convergence in total variation distance imply weak convergence?

From the definition of total variation distance, we know that convergence in total variation implies weak convergence. However, suppose we have the following, $$ d_{TV}(X_n, X) = Y_n, $$ and $\mathbb{...
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Closed form solution of the recurrence $a_{n+1} = (1-\lambda_n) a_n + \lambda_n b_n$ for $a_n$

Let $a_n, b_n$ be sequences of non-negative numbers, $n \geq 0$. Consider the recurrence: $$a_{n+1} = (1-\lambda_n) a_n + \lambda_n b_n$$ where $\lambda_n$ is some non-negative sequence. Is there a ...
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Accepting the Answer of an Optimization Algorithm that has not Converged

Is there a popular consensus on accepting the solution from an optimization algorithm that has not converged? Suppose you let an optimization algorithm (e.g. gradient descent) run for many hours, and ...
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Can we re-write Newton's Binomial formula as a polynomial in $\ r\ $ without any problems?

Newton's Generalised Binomial theorem states that if $\ x\ $ and $\ y\ $ are real numbers with $\ \vert x \vert > \vert y \vert\ (\text{note that } \left\vert \frac{y}{x} \right\vert < 1),\ $ ...
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Prove that the series $\sum^{\infty}_{n=1} n^{\alpha}(\sqrt[n]{3} - 1)$ is convergent if and only if $a < 0$.

I am trying to prove that $\sum^{\infty}_{n=1} n^{\alpha}(\sqrt[n]{3} - 1)$ is convergent if and only if $a < 0$ for some time now, but so far I did not come up with anything smart. Would you ...
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Convergent incremental recurrent sequence that does not involve \(n\)

We need a way to update a value by increasing it in a way that it converges. However, it should be done without keeping a counter of the number of operations, so $n$ should not be involved in the ...
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Show that no sequence in $E=\{\chi_A : A \subset \Bbb R \text{ discrete}\}$ will converge to $\chi_{\Bbb R}$.

Let $X=\{0,1\}^{\Bbb R}$ and each $\{0,1\}$ discrete. We can express $X$ as characteristic functions as follows $X=\{\chi_A \mid A \subset \Bbb R\}$. Show that no sequence in $E=\{\chi_A : A \subset \...
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For which $a \in \mathbb{R}$ $\sum^{\infty}_{n=1} \dfrac{(-1)^n}{n^a}$ converges and for which $a$ it converges absolutely

as mentioned in the title I want to find the values of $a$ for which series $\sum^{\infty}_{n=1} \dfrac{(-1)^n}{n^a}$ is convergent and for which values of $a$ it is absolutely convergent. Is my ...
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Convergence of the sequence of maxima of a converging sequence of functions

We are given a sequence of functions $f_n : \mathbb{R} \to \mathbb{R}$ that converge pointwise to a limiting function $f$. Each $f_n$ has a unique maximum, $x_n$, and $f$ has a unique maximum $x_0$. ...
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Prove $\sum^{\infty}_{n=1} n^{13} q^{n}$ where $q \in \mathbb{R}$ converges if and only if $q \in (-1, 1)$ using ratio test

I am trying to prove the statement above and I have found this thread in which one of the comments suggested to use the ratio test. I have come out with following solution. Is it alright? $$\lim_{n\to\...
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Convergence $\ln\left(1+\frac{1}{n^2}\right)$ [duplicate]

I´m asking for some help, since I am stuck with a proof for convergence like follows: $$\sum\limits_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$$ I tried to separate it: $$\sum\limits_{n=1}^\infty\ln\...
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Determine the serie $\sum\limits_{n=1}^{\infty}\frac{n^2+3\sqrt[]{n}}{\sqrt[]{n^5+3}}$ div or conv? [closed]

$\sum\limits_{n=1}^{\infty}\frac{n^2+3\sqrt[]{n}}{\sqrt[]{n^5+3}}$ i cant solve this serie, i tried almost all methods and I am convince that is using a comparative method, but I try it and cant solve ...
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Finding much smaller series $c_n$ than a non-increasing $a_n$ , with divergent sum, whose sum is also divergent.

While proving a theorem, the Professor stated the following lemma without proof: Let $a_n$ be a non increasing positive series, such that $\sum_n a_n=\infty$ One can find a positive series $c_n$ such ...
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Convergence of a sequence with parameter

We have a sequence defined as : Let $\lambda \in ]0, \frac{3}{4}], u_0\in ]0,1[$ and $\forall n\in N, u_{n+1}=1-\lambda (u_n)^2$. We have to study the convergence of this sequence. What I have done so ...
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convergence and inequality.

Is it necessary for the r.h.s. of the inequality to be 1 / 2 sqrt(n) ? can it also just be 1 / sqrt(n)? if so, could you please explain why.
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Is $\sum \frac{1}{n}\sin^{2}\frac{1}{n}$ divergent? [duplicate]

The question asks which one of the following is divergent? (a)$\sum_{n=1}^{\infty}\frac{1}{n}\sin^{2}\frac{1}{n}$ (b)$\sum_{n=1}^{\infty}\frac{1}{n}\log n$ (c)$\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin \...
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59 views

$\sum a_n\text{ converges}\implies\lim_{n\to\infty}\left(a_0 a_n+a_1 a_{n-1}+a_{\lfloor\frac{n}{2}\rfloor}a_{\lceil\frac{n}{2}\rceil}\right)=\ ?$

Does $\ \displaystyle \sum_{n\geq 0} a_n\ \text{ converges}\ \implies\ \lim_{n\to\infty}\left( a_0 a_n + a_1 a_{n-1} + a_2 a_{n-2} + a_{ \big\lfloor \frac{n}{2} \big\rfloor } a_{ \big\lceil \frac{n}{...
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Show $\sum_{n=2}^{\infty}\frac{1}{n(\log n)^{p}}$ is convergent.

I know $\sum_{2}^{\infty}\frac{1}{n(\log n)^{p}}$ is convergent for $p >1$ and divergent for $p\le 1$. My question is how can we show this. can we use ratio test to show this is convergent for $p &...
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Show, that if $f_n \rightarrow f$ and $f_n \rightarrow g$ is $\mu$-convergent, then $f=g$ almost everywhere on $X$

Show, that if $f_n \rightarrow f$ and $f_n \rightarrow g$ is $\mu$-convergent, then $f=g$ almost everywhere on $X$ Hint Use the fact, that: $$\left\{ x \in X \: : \: f(x) \neq g(x) \right\} = \bigcup_{...
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Interpretation of "Noise" in Function Optimization

I am trying to better understand the meaning of "noise" with regards to function optimization - specifically, why "Noisy" functions are more difficult to optimize compared to "...
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Proving Cauchy sequence and its convergence [duplicate]

As I am having trouble in concluding the question, please look at this also: For a sequence $$1 - \frac{1}{2} +\frac{1}{3}+\cdots +\frac{(−1)^{n−1}}{n} \\ \text{Then}\\ |s_n−s_m| = \left|\frac{(−1)^{m−...
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Sample covariance matrix in high dimension

I was reading this question https://stats.stackexchange.com/questions/536098/convergence-of-sample-covariance-matrix-in-case-sample-size-depends-on-dimesion and it seems that under the settings $p$ ...
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40 views

Bolzano-Weierstrass in Metric Space

I was wondering whether the Bolzano-Weierstrass Theorem ("in a finite-dimensional normed space, every bounded sequence has a converging subsequence") would hold on a finite dimensional ...
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1answer
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Use of Cauchy general principle of convergence [duplicate]

Question: Prove the convergence of sequence using Cauchy general principle of convergence. $$\frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + . . + \frac{1}{n!}.$$ My attempt: $$ | s_2 - s_1 | = \frac{1}{2!...
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Can two sequences of r.v be asymptotically equivalent?

I wanted to ask if the following notations make sense and/or are used. Convergence in distribution of a sequence $X_n$ of real random variables to the random variable $X$ is often indicated like this: ...
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Problem statement says it's a density function but I find it diverges

I'm doing a test from previous years of my school and one of the problems goes as follows; Let $(X,Y)$ be a couple of random variables with density $f_{X,Y}$ defined as : $$\forall (x,y)\in\mathbb{R}^...
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Is there a such thing called the "Classic Convergence Theorem"?

Has anyone heard of the "classical convergence theorem" before? I tried searching online for this theorem but couldn't really find anything. My Question: Does this theorem have a different ...
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Theoretically speaking, does the following algorithm make any claims on convergence for non-convex and noisy functions?

I was looking at the Robbins-Monro Algorithm, supposedly one of the original algorithms used for stochastic optimization (i.e. Some argue that the Robbins-Monro Algorithm can be considered as the ...
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What is m/n as n tends to infinity?

Here m is in twin prime pair:(6m-1, 6m+1) and n is nth twin prime-pair. I am just interested to know lower bound of difference of consecutive first twin primes as n tends to infinity. For example in (...
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Convergence of $\sum \frac{(n-1)!e^{n-1}}{n^{n-1}}$

The series given is: $$1 + \frac{1! x}{2} + \frac{2! x^2}{3^2} + \frac{3! x^3}{4^3} + ...$$ The general term of the series is given by: $u_n = \frac{(n-1)!x^{n-1}}{n^{n-1}}$ To check the series' ...
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Convergence of sequence and series [duplicate]

Show that if $a_n$ is a convergent series of non-negative reals such that {$a_n$} is decreasing for all n, then the sequence $na_n$ converges to 0. My attempt: If series $a_n$ converges, then root ...
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Newton-Raphson method convergence criteria for a system of two equation with two unknowns

I am having a system of two non-linear equations and I want to compute the root. For example the system is $$f_1(x,y)=0 \tag{1}$$ $$f_2(x,y)=0.\tag{2}$$ In this post, it is shown that the root of the ...
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Can someone please explain the significance of the subscript 2 on this picture?

[] Does the subscript here refer to the L2 norm? In the right hand side, are they saying that this is are squaring the square root of the difference in squares? Is convergence here reffering to the ...
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If sequence $2^n * X_n$ converges can we say anything about the sequence $2^{(2n)} * X_n^2$?

We know that the sequence $2^n * X_n$, ($X_n \geq 0$ for all $n$) converges to some positive constant $c$. Can we say anything about whether or not $2^{(2n)} * X_n^2$ will converge and if yes to what? ...
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Show divergence of $f_k(x):=\sin(kx)$

Show divergence of $f_k(x):=\sin(kx)$, where $x\neq m\pi$ an $m\in\mathbb{Z}$. A quick and elegant approach would be to assume (pointwise) convergence of $(f_k)$ and to notice that this would imply ...
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22 views

limit comparison test on a general series

I was given this sum: $\sum_{n=1}^{\infty}(-1)^n \sin(1/n)$, I need to determine it's convergence, First of all $$\frac{\sin(1/n)}{1/n} \rightarrow1$$ therefore the series diverges with $1/n$, then it ...
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Another proof of the divergence of $\int_1^\infty\!\frac{\ln{f(x)}}{x^2}\mathrm{d}x$, and the asymptotic behaviour of its integrand?

Reportedly, the last problem of this contest is: Given a positive decreasing sequence $\displaystyle{\{a_n\}}_{n\geqslant1}$ satisfying $\displaystyle\lim_{n\to\infty}a_n=0$, show that if the series $...
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40 views

Solving an improper integral using another

This was an old two part exam question that I was looking over. Essentially using the improper integral $\displaystyle\int_1^9 \frac{1}{\sqrt[3]{x-1}}dx$ you are supposed to determine if the integral ...
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Convergence in L2 and Uniform Boundness [closed]

I was reading about the following algorithm (https://en.wikipedia.org/wiki/Stochastic_approximation#Robbins%E2%80%93Monro_algorithm): I am trying to understand what is meant here by L2 Convergence ...
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33 views

How to use the Integral test for convergence on finite domains ($\int_0^{1} cot(\frac{\pi x}{4}) dx$)?

When using The Integral test for convergence on infinite domains $\int_a^{\infty} f(x)dx$ , the Integral test is defined as (assuming $f(x),g(x)$ are non-negative) $ \lim \limits_{ x \to \infty} \...
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109 views

$X \geq 0$, $E[X] =\infty$. What's a lower bound for speed at which sample mean $\overline X_n$ diverges toward $\infty$?

Question Let $X_1, X_2, \dots \sim F$ be an i.i.d. sequence of observations such that $X_1 \geq 0$ and $E X_1 = \infty$. Set the sample mean $\overline X_n := \frac 1n \sum_{i=1}^n X_i$. We know that $...

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