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
-1
votes
4 answers

How to prоve the convergence of the sequence?

I need to prove the convergence of $$x_n=\sum_{k=1}^{n}\frac{1}{\sqrt{k}}-2\sqrt{n+1}$$ I`m not very strong in this section of math. What should I do? Please help.
PashaMinigun
  • 175
-1
votes
2 answers

convergence of $\sum_{n=1}^{\infty} \frac{a^n}{n!}$

How can i check convergence of $$\sum_{n=1}^{\infty} \frac{a^n}{n!}$$ ? I tried some of the tests for checking if it's convergent , but it's does not work.
user510010
-1
votes
2 answers

Do a convergence test for the following series?

$$ 1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} - ....... ∞ $$ I'm able to construct its general term which is :- $$ u_n = \left( \frac{-1}{3} \right)^{n-1} $$ But I'm not sure what to do next. Any help would be highly…
underdog_eagle
  • 19
-1
votes
1 answer

Define convergence of series (criterion of Weierstrass)

Using the criterion of Weierstrass define convergence of series: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}n!}{n^{2n} }\cos2nx$$ $x\in R$
T.Wrescks
  • 95
-1
votes
2 answers

convergence in mean when mean is the constant?

For a random sequenc $X_n$, if its expectation $E|X_n|=0$, does that mean it converges in mean to $0$? For convergence in mean to $0$ we need $E|X_n-0|\rightarrow0$
martingale
  • 103
-1
votes
1 answer

Proof for a property of convergent series

Would someone be so kind as to demonstrate the proof for me? edit: $$\sum_{n=v+1}^\infty a_n=a_{v+1}+a_{v+2}+\cdots$$ The $N^\text{th}$ partial sum is: $$\begin{align}\sum_{v+1}^{v+N} a_{n}&=a_{v+1}+a_{v+2}+\cdots+a_{v+N} \\&=…
guest
  • 387
-2
votes
3 answers

Does $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge?

Sorry for the oversimplified question, but does the series $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge? The ratio test of it gives the result of "1". Thanks a lot.
darkchampionz
  • 510
  • 4
  • 13
-2
votes
1 answer

Show: if $\sum_{n>0} f(n)$ is convergent, then $\sum_{n>0} n^{1/n}f(n)$ is convergent

If $\displaystyle{\sum_{n>0} f(n)}$ is convergent, then show that $\displaystyle{\sum_{n>0} n^{1/n}f(n)}$ is convergent . I am trying using Abel's test, but I can't find my way.
Siddhanta Parial
  • 1
-2
votes
1 answer

For which parameter this kind of series converges

The sum is the following: $\sum_{k=1}^{\infty}\frac{a}{k^b}$ For which parameters this serie converges(a and b)?
user587779
  • 21
  • 4
-2
votes
2 answers

Sum of the series $\frac{1}{\sqrt n}(1+\frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n})$

I have to find the sum of the series $$a_n=\frac{1}{\sqrt n}(1+\frac{1}{\sqrt 2}+\cdots+\frac{1}{\sqrt n})$$ where $n$ tends to infinity. This series can be simplified as $$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$ where $n$ tends to…
Sona23
  • 1
-3
votes
2 answers

About Convergence Status Problem

Can I get help for this question? I have no idea. $$\sum_{n=1}^\infty = \frac {(n^n)}{(n)!n^n}$$ The problem wants convergence status from me.
Götebakar
  • 1
-4
votes
1 answer

Convergence/Divergence of Sum

How can I check if the following sum converges/diverges? I don't believe the comparison test will give me the result in this specific case. $\sum_{k=1}^{\infty}\sin (\frac{k^{2}+1}{k}\pi )$
Jacob Larsson
  • 11
1 2 3
…
35
36