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
2
votes
1 answer

Uniform convergence...

Consider $f_n(x)=n^2xe^{-nx^2}$ on $[1,\infty)$. The exercise asks to prove that $f_n$ converges uniformly. However it seems not to converge uniformly. Since if it were then it would converge to $0$ (because it converges pointwise to $0$). On the…
user16015
  • 1,020
2
votes
2 answers

Convergence iff distance equals zero

The following excerpt has been taken from Rudin's Principles of Mathematical Analysis: If $d(p_n,p) < \epsilon$ for all $\epsilon$ than $d(p_n,p) = 0$. Why isn't $d(p_n,p) = 0$ written in the textbook rather than $d(p_n,p) < \epsilon$?
Incognito
  • 1,195
2
votes
3 answers

Prove that $\sum\frac{(\log n)^2}{n^3}$ converges

This question is from Serge Lang's textbook, in a chapter that comes before the ratio and integral tests are introduced, so those can't be used. I've already proved that $\sum\frac{\log n}{n^3}$ converges and have an inkling that this result may be…
bard
  • 107
2
votes
3 answers

Convergence of $\sum_1^\infty \ln (\frac{3+n^p}{2+n^p})$

I am able to prove divergence for $p<0$ or $p=0$. How can I prove convergence/divergence for $p>0$.
Michael
  • 369
  • 1
  • 10
2
votes
2 answers

Is this a valid step in a convergence proof?

I'm asked to say what the following limit is, and then prove it using the definition of convergence. $\lim_{n\rightarrow\infty}$$\dfrac{3n^2+1}{4n^2+n+2}$. Is it valid to say that the sequence behaves like $\dfrac{3n^2}{4n^2}$ for large n?
Ldog327
  • 519
2
votes
2 answers

The Convergence of an alternating series test

Can I confirm that $$\sum \frac{(-5)^{n}}{n^{3}}$$ converges by the alternating series test?
guest
  • 387
2
votes
3 answers

Convergence of $\sum_n \frac{n!}{n^n}$

I'm working on a problem sheet and it ask to discuss the convergence of $$\sum \frac{n!}{{n}^{n}}$$ By D'Lembert's ratio test, $$\lim_{n->\infty}\frac{{a}_{n+1}}{{a}_{n}} = 1$$ and so, is inconclusive. Using Cauchy's root…
guest
  • 387
2
votes
1 answer

need radius of convergence of $e^{-x^{2}}$

I am having difficulty finding the radius of convergence of $e^{-x^{2}}$ this is for introductory analysis course. Have looked at even and odd subsequences of powerseries, but so far unable to put the pieces together. Any help…
Jebus
  • 21
2
votes
1 answer

On existence of a convergent subsequence

Let $(a_{(m,n)})_{m,n \in \mathbb{N}}$ be a double sequence of positive numbers. Suppose that we know that there exists a limit $\lim_{m \to \infty}\lim_{n \to \infty}a_{(m,n)}=L$. Does there always exists an increasing function $m: \mathbb{N} \to…
Gogi Pantsulaia
  • 511
  • 2
  • 10
2
votes
1 answer

Showing series converge by comparison

Show that the following series diverges $$\sum^{\infty}_{n=1} \sin{\left(\frac{1}{n}\right)}.$$ We do this by the comparison test. Let $a_{n} = \sin{\left(\frac{1}{n}\right)}$ Now the only test I can really apply here in my opinion is the…
user2850514
  • 3,689
2
votes
1 answer

Convergence of sums of indicator random variables

I'm working through some practice problems for my final exam and I would like to get some ideas on tackling this problem: Let $(\Omega,\mathcal{F})=(\mathbb{R}_+,\mathcal{B}(\mathbb{R}_+))$ and $\text{P}(d\omega)=\exp(-\omega)d\omega$ or…
stats134711
  • 408
1
vote
1 answer

Understanding a proof that bounded sequences in $\mathbb{R}^p$ has a convergent subsequence

I'm having trouble concerning the following proof that each bounded sequence in $\mathbb{R}^p$ has a convergent subsequence. We have already established that this is true in $\mathbb{R}$ and this is the proof that the professor gives: Use induction…
MT_
  • 19,603
  • 9
  • 40
  • 81
1
vote
1 answer

Find sum of this convergent series

find the sum of the infinite series $\sum_{i=0}^\infty\frac{2^i}{n^{(2^i)}}$ for $n>1$ I tried the following $\frac{1}{n}+\frac{2}{n^2}+\frac{4}{n^4}+...=k$ $\frac{1}{n}+\frac{2}{n^2}(1+\frac{2}{n^2}+...)=k$ But it is not helping because the…
Satvik Mashkaria
  • 3,636
  • 3
  • 19
  • 37
1
vote
1 answer

What does $ \sum_{i = 1}^{\infty} \frac{1}{i(i-1)!}$ converge to?

What does $ \sum_{i = 1}^{\infty} \frac{1}{i(i-1)!}$ converge to? That is $1 + \frac{1}{2} + \frac{1}{3*2!} + ... + \frac{1}{n(n-1)!}$
user1068636
  • 1,295
  • 12
  • 29
1
vote
1 answer

Converse of alternating series test

Is the converse of the alternating series test true? In other words, given a sequence $a_n>0$, with neither $a_{2n}$ nor $a_{2n-1}$ constant, for which there exists no positive $N$ such that $a_n>a_{n+1}$ for all $n>N$, does $\displaystyle…
Avi
  • 1,780
  • 11
  • 21
1 2 3
…
35 36