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
0
votes
1 answer

Prove a series $\sum_{n=1} ^\infty \frac{(-1)^n}{n}$ is convergent

I was asked to prove that an infinite series $$\sum_{n=1} ^\infty \frac{(-1)^n}{n}$$ is a convergent series. I tried using ratio test but the limit results in 1 which is inconclusive. I am stuck at this point. Can you give me some hint on how to…
TUC
  • 159
0
votes
2 answers

if a bounded sequence xn in [a,b] converges to x0 then x0 belongs to [a,b]

If a bounded sequence xn in [a,b] converges to x0 then prove that x0 belongs to [a,b]. I don't understand here how can x0 belong to [a,b], it may be slightly greater than b too. Please help.
Kaustav Bhattacharjee
  • 33
  • 4
0
votes
1 answer

Why does the following hold when dealing with convergence in probability

The following text is from my textbook that I have a hard time understanding. Let $X_1, X_2,X_3,...$ be independent random viariables such that $P(X_n =1) = 1 - \frac{1}{n} $ and $P(X_n = n) = \frac{1}{n}, 2 \leq n$ Clearly, $P(|X_n -1| > \epsilon)…
Carl Näsvall Sindeby
  • 35
0
votes
2 answers

When can we say that something is equal to, rather than something approaches a limit?

As an example, if we have a binary term $x$, like this $x = 0.d_1 d_2 d_3 \dots$ Where $d_1 = 1$ if $x < \frac{\pi}{ 10}$ else $d_1 = 0$ $d_2 = 1$ if $x < \frac{\pi}{ 10}$ else $d_2 = 0$ $d_3 = 1$ if $x < \frac{\pi}{ 10}$ else $d_3 = 0$ $\dots$…
Ivan Hieno
  • 503
0
votes
1 answer

Convergence of Laurent series principal part

Suppose $f(z)$ has Laurent series expansion in a neighborhood of $z=p$: $$f(z)=\sum^\infty_{k=-\infty}a_k(z-p)^k$$ A series I am interested in is $$S(z):=\sum^{-1}_{k=-\infty}a_{2k}\frac{(z-p)^{2k+1}}{2k+1}$$ Is it true that, for a fixed real…
Szeto
  • 11,159
0
votes
1 answer

convergence of Cauchy sequences defined recursively

$X=(x_n)$ defined as $X_1=1,X_2=2$ and $X_n=\dfrac12\left(X_{n-2}+X_{n-1}\right)$. To finds its limit, first we should prove that it is convergent, but my book has not given its solution. Instead, it just states that it is bounded between $1$ and…
Vivek
  • 1
0
votes
0 answers

Series convergent function series

Are there any conditions for $$\sum_{n=1}^{\infty} {\frac{x^n}{n^2}}$$ to absolutely converge to some $f(x)?$ If so, what differential equation generates this function? In general is absolute convergence in series always a consequence of some…
Narasimham
  • 40,495
0
votes
2 answers

Is my series convergence test valid?

Whilst reading through and attempting some other questions here on Math.SE, I hit upon an idea for testing whether a series is convergent, and I'd like to know: If it is a valid test How practical this test may be to use? I'll summarise it briefly…
Rhys Hughes
  • 12,842
0
votes
4 answers

Limit of $\left(1-\frac{c f(n)}{n}\right)^n$

I know that $(1-1/n)^n \approx 1/e$, but what is the result for $\left(1-\frac{c f(n)}{n}\right)^n$, where $c$ is an arbitrary constant and $f(n)$ is an arbitarry function of $n$? I am asking because I am having trouble understanding these…
fox
  • 679
  • 7
  • 18
0
votes
4 answers

Convergence test of $\sum_{n=1}^{\infty} \frac{1}{{n}^{\frac{n+1}{n}}} $

I need to test for convergence $\sum_{n=1}^{\infty} \frac{1}{{n}^{\frac{n+1}{n}}} $. I can guess that it's probably a problem for the comparison test, although I have no idea what to compare it with. All my tries have failed. Thank you for your…
Tomáš Svoboda
  • 153
  • 5
0
votes
0 answers

Which type of convergence implies that in each iteration we are closer?

Please forgive me if it is a simple answer or if I didn't get something completely, because I don't understand various types of convergence clearly. In a theorem, I can prove convergence in probability that say $\hat{\theta}_n \rightarrow \theta$ as…
user85361
  • 845
0
votes
2 answers

Finding the sum of a series that converges

I have a series $$\sum\limits_{n=1}^{\infty}\frac{1}{n^3+6n^2+8n}$$ I know the series converges because $\frac{1}{n^3+6n^2+8n}$ is less than or equal to $1/n^3$. Since $p=3$ which is greater than $1$, I know that $1/n^3$ converges. Since that…
math875
  • 11
0
votes
3 answers

How do I prove the following?

The following questions showed up in my high school math textbook and I am unsure how to approach it. Considering the sequence of partial sums {Sn} given by $Sn = \sum_{k=1}^n \frac{1}{k}$ a) Show that for all positive integers n $S_{2n} \ge S_n +…
5Arbiter
  • 9
  • 3
0
votes
1 answer

Convergence of sin(log x)

I have to show that the series $ {x_n} = { \sin (\log \ n ) }$ diverges. If I start of with $ x_n - x_{n-1} $ that doesn't get me anywhere. It is fairly obvious that $ \log n $ diverges. Can I use that and the fact that $ \sin x$ does not converge…
jeelani
  • 1
0
votes
1 answer

On the convergence of $\sum\left(\sqrt{n+1}-\sqrt{n}\right)^p$

I have to study the values of $p\in\mathbb{R}$ such that $$\sum_{n\geq 0}\left(\sqrt{n+1}-\sqrt{n}\right)^p$$ is a convergent series. So far I have found that the main term is equivalent to $n^{-p/2}$, that way $p$ should be $>2$ in order to ensure…
Rose
  • 21
1 2 3
…
35 36