Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [convergence-divergence]

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

This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.

Formally, a sequence $S_n$ converges to the limit $S.$

$$\lim_{n\to \infty}S_n=S $$ if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.

A sequence diverges if it does not converge.

For more specialized questions about divergent series, such as summation methods, consider the tag .

21990 questions
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Checking diagram of implications between modes of convergence

I learned about the definitions of these convergences. Is my diagram correct?
Danny_Kim
  • 3,423
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3 answers

What's $\lim\limits_{n\to\infty} \frac{x^n}{y^n + 1}$

For $x,y \in \mathbb{R}$, x > 1, y >0 what's the limit of $\frac{x^n}{y^n + 1}$? If $x = y \implies x^n = y^n: $ Let $\varepsilon > 0, y^N > \frac{1}{\varepsilon}, N \in \mathbb{N}$ $|\frac{x^n}{y^n + 1} - 1| = |\frac{x^n}{y^n + 1} -…
Clayton Louden
  • 155
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2 answers

Covergence of the integral $\int_0^1 \ln(1+\frac{a^2}{x^2})dx $

$\int_0^1 \ln(1+\frac{a^2}{x^2})dx $ How should i prove that this integral converges? This is what i did: $u=\frac{1}{x}$ and $dx=-\frac{1}{u^2}du$ $\int_0^1 \ln(1+\frac{a^2}{x^2})dx=\int_{\infty}^{1}…
Valentin
  • 159
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2 answers

Ratio Test Interesting Issue

I was hoping someone could help me with this interesting situation that came up while I was teaching intervals of convergence today using the ratio test. The problem asked to find the radius and interval of convergence of the series: $$…
Mada H
  • 31
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Product over a factorial convergences

I'm working on a problem and need this to converge to any value: $\displaystyle \frac{(1/2)((1/2)-1)((1/2)-2)\cdots ((1/2)-n+1)}{(n+1)!} = \Pi_{j=1}^{n} \frac{\frac{1}{2} - j+1}{j+1}$ The convergence is evident when you begin doing the product.…
Grizzly0111
  • 509
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3 answers

Proving the sum of the reciprocals squared converges

I'm investigating the Basel Problem, and the sum to consider is: $\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}$ How can I show this converges? Using graphs/computer software is also fine, but how would I do it? Is there a way using…
Jim
  • 1,210
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Does Lp convergence for absolutely continuous function imply convergence a.e.

My question is the following. If a sequence of absolutely continuos functions $\{f_n\}$ converge to zero in $L^p(S)$ ($S$ has finite measure), does it follow that the $f_n \to 0$ everwhere ?
yasin
  • 21
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3 answers

What can we say about the convergence of this series

$$\sum {z^n\over n!} $$ I used Alembert's Ration test and get $$\lim_{n \infty}{u_n\over u_{n+1}}={n+1\over z}$$ As this tends to $\infty>1$ can i say that the given series is convergent for all values of $z$ ? Note : z is a complex number
Aman Mittal
  • 2,091
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Divergence of $\sum \frac{a_j}{1+a_j}$

Given: $a_j >0$ and $\sum a_j$ diverges. Show that $\sum \frac{a_j}{1+a_j}$ diverges. Hint: show that if it converged, $a_j$ -> 0. I don't understand how to think about this problem. Is there a convergence test I should use? I tried starting with…
kiwifruit
  • 707
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2 answers

Conditions for convergence

Let f be a continuous function on [0,1], and $f_n(x)=f(x)^n$. Under what conditions on f will the sequence converge point wise? Uniformly? I think it will converge uniformly if $-1\leq x \leq 1$ but I am not sure about point wise.
user138602
  • 39
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1 answer

Sequence convergence proof

Sequence: $a_n=5r^{n-1}$ and $|r| \lt 1$ I am finding an $N$ for a convergence proof. My result is $$N \ge \frac{\ln\varepsilon/5}{\ln r} +1$$ And the textbook answer is $$ N \ge \frac{\ln 5/ \varepsilon}{\ln 1/r} +1 $$ Are these answers…
user120494
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0 answers

Convergence of $\sum_n^\infty (-1)^n\frac{\sin^2 n}n$ , questions

Could anyone give a hint how to prove the convergence of the following sum? $$\sum_n^\infty (-1)^n\frac{\sin^2 n}n$$ I tried writing it like this instead: $$\begin{align}\sum_{n=1}^N (-1)^n\sin^2 n &= \sum_{n=1}^N (-1)^{n+1}…
user103670
  • 21
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3 answers

Determine convergence $\sum_{n=1}^\infty\left(\frac{2 n + 2}{2 n + 3}\right)^{n^2}$

Does $$\sum_{n=1}^\infty\left(\frac{2 n + 2}{2 n + 3}\right)^{n^2}$$ converge? Hi, I was wondering if anyone knows how to solve this problem? I think I can't use root test... because the result is 1 and it is meaningless. Thank you guys very much.
asaak
  • 65
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4 answers

Series Convergence Criterion

The greatest value of p such that the series \begin{equation} \sum_{n=1}^{\infty}(-1)^n \cdot \tan \left(\frac{1}{\sqrt{n^p}}\right) \cdot \ln \left(1+\frac{1}{n^{2 p}}\right) \end{equation} converges conditionally, what is the value of p?
Thermal
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5 answers

Definition of convergent, can the N change if I change to another $\epsilon$

I’m having problems with the definition of converging sequence. In my lecture note, the definition is: a real sequence $a_n$ converges to the limit a if for all $\epsilon>0$ there exist a natural number N such that $|a_n-a|<\epsilon$ for all n>N I…
Eileen
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