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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34 views

If all subsequences $\{x_{n_i}\}$ of $\{x_n\}$ $ \lim_k\frac1k\sum_{i=1}^{k} {x_{n_i}}= y $ then $\lim_n x_n= y$

Let $\{x_n\}_n$ be a real sequence and $y\in\mathbb{R}$ such that for all subsequences $\{x_{n_i}\}$ of $\{x_n\}$ we have $$ \lim_k\frac{1}{k}\sum_{i=1}^{k} {x_{n_i}}= y $$ My problem: Why $\lim_n ...
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Show that the sequence $(a_n)$ approaching the common arithmetic-geometric mean from $a_0,b_0$ from bellow and $(b_n)$ approaching..

Let $0<a_0<b_0$ and $a_1:=\sqrt{a_0b_0},\,\,b_1:=\frac{a_0+b_0}{2}$ and $a_2:=\sqrt{a_1b_1},b_1:=\frac{a_1+b_1}{2}$ in genereal: $a_{n+1}:=\sqrt{a_nb_n}$ and $b_{n+1}:=\frac{a_n+b_n}{2}$ ...
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1answer
19 views

Convergence of an infinite series that contains a converging series

I think that the series $$\alpha\beta\gamma+\alpha^{2}\beta\left(\gamma+\gamma^{2}\right)+\alpha^{3}\beta\left(\gamma+\gamma^{2}+\gamma^{3}\right)+\dots+\alpha^{n}\beta\left(\gamma+\gamma^{2}+\dots+\...
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Limit/convergence of a series of two shifted error functions

I try to find the limit (or at least the proof for convergence) of the infinite sum $$\sum_{\substack{k=-\infty \\ k \neq 0}}^{\infty} e^{-\frac{a}{b}ck} \left| \text{erf}\left( \frac{a + i(b-ck)}{\...
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2answers
75 views

$1/(1-x)$ series: dividing by zero?

I am reading a book "A History of Mathematics by Boyer" In the chapter about Euler it states that "Although on occasion he warned against the risk in working with divergent series, he himself used ...
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1answer
24 views

An Understanding of Dominated Convergence theorem

I have read that The Dominated Convergence Theorem: If $\left\{f_{n}: \mathbb{R} \rightarrow \mathbb{R}\right\}$ is a sequence of measurable functions which converge pointwise almost everywhere to $f,$...
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1answer
18 views

A type of set used in convergence in measure theory

This is not a specific problem, but a general question. Often when we're showing convergence of functions (particularly pointwise) or even of sets in certain cases, a set of the following form ...
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Conditions when morphism saves convergence

If $A_N\to A$ on $R$ in distribution, and $\xi :\ R\to S^1$, when is it true that $\xi(A_N)\to \xi(A)$?
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What does the Kalman filter generally converge to? And why?

So, i'm guessing whomever shows up here, knows what the Kalman filter is. It's quite an extensive model to type out, so here is an explanation from MIT (see ch. 11.5) We have a feeling that it ...
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“generalised” gamma-like integral $\int_0^\infty x^ne^{-f(n)x}dx$

I have noticed that if we have an integral of the form: $$I[f]=\int_0^\infty x^ne^{-f(n)x}dx=\frac{1}{f^{n+1}(n)}\int_0^\infty x^ne^{-x}dx=\frac{n!}{f^{n+1}(n)}$$ I was wondering what kind of ...
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1answer
24 views

Convergence of generalised continued fractions (with positive partial numerators)

Suppose that we have a sequence of positive numbers $(x_n)_{n \in \mathbb N}: x_n>0$ which are not necessarily integers. Q1 Can you give some examples of necessary/sufficient conditions for the ...
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2answers
96 views

Is $\int_E \frac{1}{(x^2+y^2)^2}dxdy$ convergent?

I have to tell whether this integral is convergent: $$\int_E \frac{1}{(x^2+y^2)^2}dxdy$$ where $E=\{0\leq y \leq x^a\} \cap \{x^2+y^2\leq 1\} $. I'm asked for which $a \geq 0$ the integral converges. ...
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find the value of instablity error from which this value shows instability

I have an Euler method that has this form: $$\hat{I}(t_{n+1}) = \hat{I}(t_{n})+h\beta \hat{I}(t_{n})(1-\frac{\hat {I}(t_{n})}{N})$$ which can also be written like $$\hat{I}(t_{n+1})=\phi (\hat{I}(t_{...
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44 views

Even and odd functions for Taylor serie

I have asked this question before, sorry, but I'm still confused about how I can show it. Hope anybody can help me? We let $f:\mathbb{R}\to\mathbb{R}$ be infinitely often differentiable function and ...
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57 views

Does the series $\sum_{n=4}^{\infty} \frac{\sqrt{1+\frac{1}{n}}}{n^{2}}$ converge? [closed]

Study the convergence for the following series: $$\sum_{n=4}^{\infty} \frac{\sqrt{1+\frac{1}{n}}}{n^{2}}$$
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Prove that if $E[X_1^p]<\infty$, then $\frac{\max_{1\le i\le n} X_i}{n^{1/p}} \rightarrow 0$ in probability where $\{X_n\}$ is i.i.d and non-negative

Suppose $\{X_n\}_{n\geq 1}$ are iid and non negative. Define $M_n=\max \limits_{i=1,\ldots,n}\{X_i\}.$ Prove if $E[X_1^p]<\infty$, then $\frac{M_n}{n^{1/p}}\rightarrow 0$ in probability. As a ...
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2answers
50 views

Why is this sequence not convergent? [duplicate]

Definition of a convergent sequence: A sequence ($a_n$) converges to a real number $a$ if, for every positive real number $\epsilon$, there exists an $N\in\mathbb{N}$ such that whenever $n>N$...
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1answer
43 views

Prove convergence in distribution of sum of non-i.i.d random variables.

$X_1,X_2,X_3 ,\ldots$ be independent random variables with distribution $P(X_i=i)=P(X_i=-i)=1/2$ for all $i$. Define $S_n=X_1+X_2+X_3+\cdots+X_n$. And the question is to show "Does $\{S_n/n^p\}_{n=1}^...
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40 views

Taylor serie for even function. Proof [duplicate]

We let $f:\mathbb{R}\to\mathbb{R}$ be infinitely often differentiable function and we let the Taylor series be: $$\displaystyle\sum_{n=0}^{\infty}\left(\left(\frac{f^{n}(0)}{n!}\right)x^n\right) $$Let ...
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1answer
70 views

Show that the series $\sum_{n=1}^\infty \sin \left( \frac{x}{n^2} \right)$ does not converge uniformly

I asked this question about a week ago but I am little bit unsure about the way to solve it so I hope it is ok if I ask again about some things I do not fully understand. I have to show that the ...
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62 views

Does $\sum \limits_{k=1}^{\infty} \frac{\cos(k)-\cos(k+1)}{k}$ converge?

I was given the following series and I'm asked to decide (and prove) whether it converges or diverge: $\displaystyle\sum \limits_{k=1}^{\infty} \frac{\cos(k)-\cos(k+1)}{k}$ So far, I couldn't ...
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1answer
44 views

Uniform convergence and integrals.

I'm asked to tell if the following integral is finite: $$\int_0^1 \left(\sum_{n=1}^{\infty}\sin\left(\frac{1}{n}\right)x^n \right)dx$$ I studied the series (which converges uniformly on $(-1,1)$ by d'...
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4answers
53 views

A closed discrete set

Let $V$ be a normed vector space. Let $(b_n)\subseteq V, b_n \to b\in V.$ Show that $B := \{b,b_1,b_2\dots\}$ is closed. I know that if $b_n\to b,$ then $b_n$ is Cauchy. That is, $\forall \...
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1answer
26 views

How to prove the sequence $ \frac{c^n}{\sqrt{n}}, c \in (0, 1)$ is convergent

Given a sequence $a_n = \frac{c^n}{\sqrt{n}}$ where $c \in (0, 1), n = 1, 2, 3, \cdots$, how to prove that the sequence is convergent? What if $c \in (0, \infty)$?
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1answer
84 views

Determining whether $\int_{1}^{+\infty}\frac{\sin^3 \left(x\right)}{\sqrt {x^2}}\,\mathrm{d}x $ converges or diverges.

I's struggling with an integral, and not sure wich method I should use to determining whether it converges or diverges. I know, from a software, that it should converge. The integral is: $$ \...
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2answers
40 views

Convergence of series $\sum_{k=2}^{\infty} \frac{2^{k}}{\lfloor{\frac{k}{2}\rfloor}}$

I would like to inspect the convergence of the following series $$\sum_{k=2}^{\infty} \frac{2^{k}}{\lfloor{\frac{k}{2}\rfloor}}$$. Because I am new to the whole series part it would be very nice if ...
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1answer
13 views

Some doubts about Levy's Continuity Theorem proof - Convergence results

THEOREM (Levy's Continuity Theorem) Let $(\mu_n)_{n\geq1}$ be a sequence of probability measures on $\mathbb{R}^d$, and let $(\hat{\mu}_n)_{n\geq1}$ denote their characteristic functions (or Fourier ...
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1answer
32 views

Radius of convergence for binomial series (2)

I'm having trouble calculating the radius of convergence for for the following binomial series. More in detail, I'm having trouble finding $c_k$ and $c_{k+1}$ for the following series: $$ \sum_{k=0}^...
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1answer
37 views

Convergent on edge. 1/a

For $a \in \mathbb{R}\setminus\{0\}$ we have function $f$:$\mathbb{R}\setminus\{a\}\rightarrow \mathbb{R}$: $$f(x)=\frac{1}{a-x}$$ for $x\in \mathbb{R}\setminus\{a\}$. Then I have to find the Taylor ...
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1answer
42 views

How To Determine If $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\sum_{k=0}^{n-1}\binom{2k}{k}\binom{k}{n-k}\right)$ Converges or Diverges?

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\sum_{k=0}^{n-1}\binom{2k}{k}\binom{k}{n-k}\right)$$ Question : How do i determine if the above Series Converges to Diverges? I have no idea where to begin ...
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2answers
73 views

Is this series convergent? $1 + 1/2 + 1/2 + 1/4 + 1/4 + 1/4 + 1/4 + …$ [closed]

Is this series convergent? How to prove? $$1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + ... + \frac{1}{8} \ (8 \times 1/8) + \frac{1}{16} + .....
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21 views

Convergence of functional series $\sum_{k=0}^{\infty}\frac{(2x)^{k}}{(1+x^{2})^{k}}$

Find convergence area and absolute convergence area of the following series: $$\sum_{k=0}^{\infty}\frac{(2x)^{k}}{(1+x^{2})^{k}}$$ I wrote it as $$\sum_{k=0}^{\infty}\frac{(2x)^{k}}{(1+x^{2})^{k}}=\...
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0answers
36 views

Rewriting $\sum_{k=0}^{\infty} e^{-xk}$

I'm trying to calculate the radius of convergence of the series: $$\sum_{k=0}^{\infty} e^{-xk}$$ In general I know how to calculate the radius of convergence, yet I seem a stuck with the idea that $x$...
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2answers
23 views

Justify Convergence of a sequence [closed]

If the sequence $\left \{ |a_{n}| \right \}$ is convergent. Is the sequence $\left \{ a_{n} \right \}$ convergent? Justify? Guys, I know this might seem like a homework question, it isn't. I'd ...
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norms are not equivalent - proof

Consider the sequence of functions $f_n : [0, 1] \to \mathbb{R}$ given by $$ f_n(x) = \begin{cases} 1 - nx &0 \le x \le 1/n\;\\ 1& \textrm{otherwise}\;\;, \end{cases} $$ ...
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1answer
37 views

Doubt in a part of the proof of Bolzano-Weierstraß Theorem

Let $n$ be a natural number and $\{a_n\}$ be a bounded sequence of real numbers, that is $\vert a_n\vert\leq M$, for all $n$ ($M\geq0$). Define $E_n=\overline{\{a_j\vert j\geq n\}}$ as the closure of ...
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0answers
24 views

Convergence proof on Newton's method from Boyd & Vandenberghe's book on convex optimization

From Boyd & Vandenberghe's book, page 491: Applying the Lipschitz condition,we have $$\begin{aligned} \left\|\nabla f\left(x^{+}\right)\right\|_{2} &=\left\|\nabla f\left(x+\Delta x_{\mathrm{...
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2answers
21 views

Almost sure convergence of random variables with same mean and the difference goes to zero on the product

Let $X_n$ be a sequence of independent real valued random variables on the same event space, with the same (finite) mean $\mu$. Suppose that for almost every couple of points $(\omega,\omega')$ in ...
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2answers
54 views

Prove by definition: $\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$ diverges [closed]

I got this question: Definition: A series $\displaystyle\sum_{n=1}^{\infty}a_{n}$ called converges if the limit $\displaystyle\lim_{k\to\infty}\sum_{n=1}^{k}a_{n}$ exists and final, otherwise the ...
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1answer
44 views
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Interval of convergence of Lagrange's infinite series

I am reading a book on Orbital Mechanics for Engineering Students by Howard D. Curtis. In that book it was mentioned (in page 119) that there is no closed form solution for $E$ as a function of the ...
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Are these sufficient conditions for convergence of this series?

If ${a_n}$ is a strictly decreasing sequence of real numbers with $\lim\limits_{n\to\infty} a_n = 0$, then does it imply that $$\sum\limits_{n=1}^\infty \left((-1)^n \cdot \frac{a_1 + a_2 + \ldots + ...
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1answer
55 views

Does $\sum\limits_{n=1}^\infty\frac{1}{n+\sqrt{n}}$ diverge? [duplicate]

I'm trying to do this: $\dfrac{1}{n+\sqrt{n}} \geq \dfrac{1}{n+n} = \dfrac{1}{2n} \Rightarrow \dfrac{1}{2n} \leq \dfrac{1}{n+\sqrt{n}} \Rightarrow \dfrac{1}{n} \leq 2\cdot \dfrac{1}{n+\sqrt{n}}$. So, ...
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31 views

Is this $a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{…^{(\frac{1}{n!}-\frac{1}{(n+1)!})}}}$ have a finit limit?

My question here is related to telescopic sum using factorial and it is related to my question here, I have computed some values of $a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{...^{(\frac{1}{n!...
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0answers
24 views

Convergence (or not) of a complex series

Determine whether the series $$\sum_{n=2}^{\infty} \frac{1}{n\ln{(n)}+\sqrt{(\ln{(n)})^3}}$$ convergence or not. I’m guess that we use the comparison test here comparing the series above to $$\sum_{n=...
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1answer
55 views

If $\{f_n\}$ is $L^1$-weakly convergent sequence then $\{f_n\}$ converges in measure?

Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ be a $L^1$-weakly convergent sequence to $f\in L^1$. Can we say that $\{f_n\}$ converges in measure to $f$?
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1answer
47 views

PDE problem on $L^2$ convergence

With $I \subset \Bbb{R}$, let $\chi_I$ denote the indicator function of I, $$\chi_I(x)=\begin{cases} 1 & \text{if x $\in$ I} \\ 0 & \text{otherwise} \end{cases}$$ For any $k \in \Bbb{N}$, ...
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1answer
26 views

Prove that the following is not uniformly convergent

previously I have proved that the following series converges uniformly in $[a,\infty) ,a>0$ $$ \sum\limits_{n=1}^\infty2^n\sin(\frac{1}{3^nx})$$ But I was requested to prove that it doesn't ...
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2answers
49 views

Find the radius of convergence for infinite series. [duplicate]

I have just shown via the Taylor expansion for $\sin(\frac{1}{n})$ that the series $$ \sum_{n=1}^{\infty}\left(\frac{1}{n} - \sin\left(\frac{1}{n}\right)\right) $$ is in fact convergent and now I'm ...
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0answers
25 views

How to prove that $\sum\limits_{n=1}^\infty2^n\sin(\frac{1}{3^nx})$ is uniformly convergent in $[a,\infty)$ [duplicate]

Prove that the following series converges uniformly in $[a,\infty) ,a>0$ $$ \sum\limits_{n=1}^\infty2^n\sin(\frac{1}{3^nx})$$ So, I need to prove that for every $x\geq0$ and every $\epsilon > ...
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2answers
54 views

Does $\sum\limits_{n=1}^\infty\frac{1}{n+\sqrt{n}}$ converge or diverge? [closed]

I know it is divergent but I don't know how to prove that it is. Thanks for help

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