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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51 views

Showing that $\sum\limits_{j=0}^\infty a_j=\sum\limits_{k=0}^\infty b_k$

Show when $\sum\limits_{j=0}^\infty a_j$ is absolute convergent, than $\sum\limits_{k=0}^\infty b_k$ with $b_k:=(a_0+2a_1+...+2^ka_k)/2^{k+1}$ is also absolute convergent, and even $\sum\limits_{j=0}^\...
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2answers
65 views

Show that $\sum\limits_{j,k=2}^\infty\frac{1}{j^k}$ converges and calculate the limit of the series

Show that $\sum\limits_{j,k=2}^\infty\frac{1}{j^k}$ converges and calculate the limit of the series. My approach: We look if one of the iterated series converges absolutly. $$\sum\limits_{j=2}^\infty\...
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1answer
30 views

Limit of a discrete convolution is equal to zero

The problem: Let $ (u_n)_{n \in \mathbb{N} } $ and $ (v_n)_{n \in \mathbb{N} } $ be two sequences of real numbers convergents to zero. Suppose that there exists a number $M>0$ such that \begin{...
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51 views

Find $\lim_{n\to\infty}((\tan(\frac{\pi}{2n})…\tan(\frac{n\pi}{2n}))^{\frac{1}{n}}$

Now, \begin{align} &\displaystyle\lim_{n\to\infty}\left(\frac{\tan(\frac{\pi}{2n})}{\frac{\pi}{2n}}\cdots\frac{\tan(\frac{n\pi}{2n})}{\frac{n\pi}{2n}}\right)^{\frac{1}{n}}\frac{(\pi) }{2n}(1.2.3......
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1answer
40 views

Convergence of the generalized Frullani's integral

After viewing some topics on MSE concerning Frullani's integral I ask for a generalized version . I think it's too hard to get the result so I ask just for the convergence : Let $f(x)$ be a ...
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15 views

Radius of convergence of this complex-numbers power series

I would like to calculate the radius of convergence of power series: f(z)=\sum{n=0}^∞ n!z^(n!) My mathematics book says that I can calculate it with Cauchy's test (limsup[n→∞] |a_n|^(1/n)). However, I ...
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42 views

How does the series $\sum\frac{x^{n+1}}{n3^{n}}$ converge when $x = -3$?

My lecturer said it converges at $x = -3$ and diverges at $x=3$. I agree it diverges at $x = 3$ because it is the series $\sum\frac{3}{n}$. But at $x = -3$, isn't it $\sum-\frac{3}{n}$ which diverges ...
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43 views

Taylor series $\log((1-z^3)/(1+z^2))$ [duplicate]

I need to find the taylor series of $$\log \frac{1-z^3}{1+z^2}$$ around $0$ and the radius of convergence. I wrote $\log((1-z^3)/(1+z^2))=\log(1-z^3)-\log(1+z^2)$ And then wrote the Taylor series for ...
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31 views

$\sigma$ is a permutation of $\mathbb{N}$,then the sum $\sum\sigma(n)/n^2$ ,$N<n<4N$

The options are : (a)$\sigma$ is only an injective function then also the sum is convergent (b) the sum is bounded above (c) the sum is bounded below (d) has 0.125 as a lower bound Now I think options ...
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4answers
45 views

Proof that $\sum \limits_{k=0}^{\infty} \left( k+1\right) \cdot \left( -x\right)^{k}$ converges

I'm asked to proof the convergence of $\sum \limits_{k=0}^{\infty} \left( k+1\right) \cdot \left( -x\right)^{k}$ to $0$ for $x\in\left( 0,1 \right)$ Well, I've started with the alternating series test:...
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2answers
29 views

Checking the convergence of the improper integral

So I have the following function to which i should check the convergence: $$\int_{1}^{+\infty} \frac{\arctan(-e^{x})}{\sin(\frac{1}x{})}$$ I thought that one way to do it was with a comparison test. ...
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25 views

Backwards direction of Cauchy Criterion for Sequences of Functions

I am reviewing the proof of the Cauchy Criterion for sequences of functions and have a question regarding the backwards direction. Statement: Let $A\subseteq \mathbb{R}$ and $(f_n)$ be a sequence of ...
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1answer
67 views

Monotone Convergence theorem Application

$$ \lim _{n \to \infty} \int_{-\infty}^{\infty} \frac{e^{-x^{2} / n}}{1+x^{2}} d x=? $$ My opinion is using Monotone Convergence Theorem here. For every $x \in \mathbb{R}$ the sequence $\left\{e^{-x^{...
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Sufficient condtions for almost sure convergence - proof verification

I want to prove the following lemma, relating to the proof of Kolmogorov's Two Series Theorem: Obviously, it suffices to show that $\{Y_k\}_{k \in \mathbb{N}}$ is Cauchy for almost every $\omega$. I'...
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1answer
27 views

If $E|f(Z)|<\infty$ for $Z\sim N(0,1)$ then is $E|f( Z/\alpha)|<\infty$ for some $0<\alpha<1$?

Suppose $Z\sim N(0,1)$. Suppose $f:\mathbb R\to\mathbb R$ is a function that is integrable w.r.t. the density of $Z$ i.e. $E|f(Z)|<\infty$, in other words $\int_{-\infty}^\infty |f(x)|e^{-x^2/2}dx&...
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2answers
53 views

determine the values p for which the limit of $a_n$ converges

I came across the problem below. $0<a_{n+1}\leq a_n+1/n^p$ where $p$ is a real number and $1\leq n$. For which values of $p$ does $<a_n>$ converge? First I thought it would look somewhat like ...
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65 views

Which of the following series is/are convergent?

Which of the following series is/are convergent? (a)$\sum_{n\ge2}\frac{1}{n\log n}$ (b)$\sum_{n\ge2}\frac{\log^2 n}{n^2}$ (c)$\sum_{n\ge2}\frac{1}{n\log^2 n}$ (d)$\sum_{n\ge2}\frac{\sqrt{n+1}-\...
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1answer
31 views

Integral test fails?

I have a problem with a series. We have the series $$\sum \limits_{n=1}^{+\infty} \frac{1}{(n+2)(n+4)}.$$ When I used the integral test for convergence (correct me if I'm wrong), the result is $$\lim_{...
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2answers
55 views

Why $\int_{ \mathbb{R}^2 } \frac{dx\,dy }{(1+x^4+y^4)} $ converges?

Why $\int_{ \mathbb{R}^2 } \frac{dx\,dy }{(1+x^4+y^4)} $ converges? Apparently this integral is quite similar to the integral $\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or ...
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1answer
40 views

given a divergent series, can we conclude a related sequence is not converging to zero?

say we have a sequence of non-negative reals, $a_1, a_2, \dots$, and that $\displaystyle\sum\limits_{n=1}^{\infty}a_n$ is divergent, meaning convergent to infinity. Under this scenario I am trying to ...
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2answers
152 views

Proving $\lim_{n\to\infty}\sup_{\lambda \in \Lambda}\big|\int_\mathbb X f_{n, \lambda}(x) \mu_n(d x)-\int_\mathbb X f_\lambda(x) \mu(d x)\big|=0$

Could any one help me to prove the following? Let $X$ be a Borel space, suppose that we have a family of uniformly bounded real Borel measurable functions $\{f_{n,\lambda}\}_{n\ge 0, \lambda\in J}$ ...
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89 views

What is $\lim_{x\to \infty } \frac{\tan x}{\log x} $?

I'm interested to evaluate this $\displaystyle \lim_{x\to \infty } \frac{\tan x}{\log x} $, But really I can't juge whether it is convergent or divergent , Wolfram alpha suggested that dosn't exist as ...
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1answer
37 views

Show that $f(x):=\sum\limits_{n=0}^{\infty}\frac{1}{n}h(2^{n}x),$ where $h$ is a piecewise function, converges uniformly on $[0,1]$

For $x\in\mathbb{R}$, consider a piecewise function defined by $$h(x):=\left\{ \begin{array}{ll} x,\ \ \ 0\leq x\leq 1\\ 2-x,\ \ 1\leq x\leq 2\\ 0,\ \ \text{otherwise}. \end{array} \right.$$ Now, ...
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6 views

graph structure affecting the convergence of power method [closed]

how does the graph structure affect the convergence of the power method? And how can you tell when your algorithm converged?
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60 views

Sequence defined $f_n = \frac{a}{1+f_{n-1}}$ is convergent?

Sequence $<f_n>$ defined by $f_n = \frac{a}{1+f_{n-1}}$ where $f_1$ and $a$ are positive. Is this sequence monotonic ? I only know, If this sequence is convergent, then it's limit will be the ...
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2answers
40 views

Prove convergence using limit comparison or direct comparison $\sum \limits_{n=1}^{+\infty} \frac{2^n}{n!}$ [closed]

Prove that the series converge using direct comparison or limit comparison $$\sum \limits_{n=1}^{+\infty} \frac{2^n}{n!}.$$ I really don't know how to proceed with the comparison tests though I know ...
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21 views

Finding what an alternating series converges to

One of my homework problems was to determine whether the series $\sum_{n=0}^\infty \frac{(-1)^n x^{2n-4}}{(2n-1)!}$ converges, and if it does, find what it converges to. I used the ratio test to show ...
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17 views

Proof of convergence for Customer Retention Rate Series

I tried to practice and see old notes of Calculus 1, but I can't still find out the reason why my series converge. Retention Rate is a ratio that defines how many customers will shop again after the ...
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1answer
46 views

Integral and sum function

I have that $$\sum_{n=-\infty}^{\infty}\left(\left(\frac{1}{1+n^2}\right)e^{inx}\right)\text{.}$$ Where $f$ is the sumfunction for the serie. Then I have to find $\int_{-\pi}^{\pi}f\,sin(x)\mathrm{d}x$...
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2answers
45 views

Showing $X_n\sim \operatorname{Bin}\left(1,\frac{1}{n}\right)$ almost surely does not converge to $0$

I want to show that $ X_n\sim \operatorname{Bin}\left(1,\frac{1}{n}\right)$ almost surely does not converge to $0$; $X_n$'s are independent. Therefore I got the hint to use Borel-Cantelli and showed ...
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37 views

Show that $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.

I am trying to prove that this infinite series $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$. I can definitely show that this series converges ...
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41 views

Does $^{\infty}i$ actually converge to $\frac{2i}{\pi}W(-\frac{\pi i}{2})$?

The question Using only the principle branch of the complex log. Skipping to the end, if $z = \frac{2i}{\pi}W(-\frac{\pi i}{2})$, which is the fixed point of $z \rightarrow i^z$, it's not true that a ...
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1answer
22 views

Vanishing trace implies convergence to zero?

Let $A_n, B_n$ be sequences of $k \times k$ matrices such that $A_n$ converges to a positive semi-definite matrix $A$ and $B_n$ converges to a positive definite matrix $B$ (w.r.t. Frobenius norm). ...
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2answers
53 views

If $q_n$ converges to $b$ ,why is $b$ part of the natural numbers in a sequence

A question I found while researching reads as follows : Let $q_n$ be a sequence in $\mathbb{N}$ . Suppose $q_n \to b$ in $\mathbb{R}$ I am trying to explain why $b \in\mathbb{N}$. Logic tells me that ...
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3answers
41 views

Which of the following is divergent? $\sum\frac1n\sin^2\frac1n$, $\sum\frac1{n^2}\sin\frac1n$, $\sum\frac1n\log n$, $\sum\frac1n\tan\frac1n$

Which of the following is divergent? (a) $\displaystyle \quad\sum_{n=1}^{\infty}\frac{1}{n}\sin^2(\frac{1}{n})$ (b) $\displaystyle \quad\sum_{n=1}^{\infty}\frac{1}{n^2}\sin(\frac{1}{n})$ (c) $\...
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2answers
62 views

$\sum_{n=1}^\infty\frac{(-1)^n\sin ^2n}{n}$ Is the following solution wrong ?; Does $\sum\frac{(-1)^n\cos 2n}{2n}$ converge?

$$\sum_{n=1}^\infty\frac{(-1)^n\sin ^2n}{n}$$ Solution from the lecture notes : $$\frac{(-1)^n\sin ^2n}{n}=\frac{(-1)^n(1-\cos > 2n)}{2n}=\frac{(-1)^n}{2n}-\frac{(-1)^n\cos 2n}{2n}$$ $\frac{(-1)^n}...
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1answer
45 views

Prove the convergence of the integral $\int_{0}^{\infty}\frac{x^n}{\Gamma\Big(\operatorname{W}(x)\Big)}dx$

Prove that the following integral is convergent $\forall \,n\geq 1$ a natural number : $$\int_{0}^{\infty}\frac{x^n}{\Gamma\Big(\operatorname{W}(x)\Big)}dx$$ Where we have the Gamma function and the ...
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1answer
33 views

Can we say $\sum_{n=1}^{\infty}\frac{2n^2-n-2}{3n^2-n-1}$ converges to $2/3$? [closed]

Based on the divergence test, it should diverge.
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50 views

What are the values of $a$ for which this integral converges?

What are the values of a for which this integral converges? $$I = \int_{0}^{\infty} \frac{\sin x}{x^a}\,dx.$$ I tried comparing it with the integral $$\int_{0}^{\infty} \frac{1}{x^a}\,dx.$$ but I ...
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1answer
67 views

Convergence in probability of $X_n\sim \operatorname{Bin}\left(1,\frac{1}{n}\right)$

Maybe you can help me with the following task. I still have a lot of problems with convergence of random variable. Let $X_n$ $(n \in \mathbb{N})$ independent random variables on $(\Omega,F,P)$ with $...
2
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1answer
54 views

What is the Solution to this sum $\sum \limits_{n=1}^{\infty}(1-(-1)^{\frac{n(n+1)}{2}})(\frac{1}{2})^n$

what is the value of this series $\sum \limits_{n=1}^{\infty}(1-(-1)^{\frac{n(n+1)}{2}})(\frac{1}{2})^n$ ? Anything what's solid and that i got so far is only $\sum \limits_{n=1}^{\infty}(1-(-1)^{\...
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2answers
39 views

If $T$ is a bounded linear map and $\sum x_n$ is an absolutely convergent series, then $T(\sum x_n) = \sum T(x_n)$

Is the following true? If so, how to prove it? If $T:X \to Y$ is a bounded linear map between the Banach space $X$ and the normed vector space $Y$ and $\sum x_n$ is an absolutely convergent series, ...
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2answers
34 views

limit of a sequence $\frac{(-1)^n}{n}$

Finding Convergence or divergence of sequence $$a_{n}=\frac{2+(-1)^n}{n}$$ What I try :: A sequence $\{a_{n}\}$ is convergent if $\displaystyle \lim_{n\rightarrow \infty}a_{n}=0.$ Otherwise it is ...
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1answer
29 views

Show that the sum function $f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)$ is continous

Consider for $x \in \mathbb{R}$ the sum function defined as $$ f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1) $$ I have shown that the series converges point wise by using that $$ |exp(-...
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0answers
23 views

Show convergence of indirectly defined series.

For $a,b\in\mathbb{R}^+$ we define $$ \alpha_i = \frac{2\pi}{2^ib}\quad \mathrm{ for }\quad i\in \mathbb N_0\qquad h_0=\frac{1}{a^2\alpha_0^3} $$ and $$ h_i=a\bigg (\alpha_i(h_i+\sum_{j=0}^{i-1}h_j) \...
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2answers
70 views

Convergence $\int_1^{\infty} x^2 \cos(e^x)\,dx$

I am learning about ways to test if an integral converges or diverges and I am stuck with this one:$$\int_1^{+\infty}x^2\cos(e^x)dx=?$$ UPD: I forgot to say that I had to explore absolute and ...
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0answers
47 views

Dirichlet series and analytic properties of w(n)

w(n)=v(n) is the number of distinct prime factors of n. i have found Its Dirichlet Series and Abscissa of Convergence from Apostol's book. Proof. Let $a_{n}$ indicate whether $n$ is prime. For $\sigma&...
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5answers
90 views

Find a convergent sequence with $\sum \limits_{n=0}^{\infty} a_n = \sum \limits_{n=0}^{\infty}a_n^2$

If $(a_n)_{n\in N_0}$ and $a_n>0$, find a convergent sequence $a_n$ with $\sum \limits_{n=0}^{\infty} a_n = \sum \limits_{n=0}^{\infty}a_n^2$ , whereas $\sum \limits_{n=0}^{\infty} a_n$ and $\sum ...
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1answer
21 views

Controlling the convergence of a series

I have a sequence of real numbers $(a_n)_{n \in \mathbb N}$ such that each $a_n$ is positive and the $a_n$s decrease monotonically with limit zero. Is there any way to control the convergence of the ...
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1answer
36 views

Prove there exists $\alpha \ge 0$ s.t $\int_0^\alpha f(x)dx =\int_0^\infty g(x)dx$ given that $f,g\ge 0$, $F(x)$ diverges and $G(x)$ converges

This is one of the problems we got as an assignment: if $f(x),g(x)$ are two integrable functions on $[0,t]$ for any $0<t\in \Bbb{R}$. and suppose that: $f(x)\ge 0,\ g(x)\ge 0$, for all $x\ge 0$ $\...