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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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Convergence of $\sum_{t=1}^{\infty}{\left(\frac{1}{2}\right)^t} \ln\left(\frac{W+2^{t-1}-P}{W}\right)$

In a paper on the St Petersburg Paradox, it is said that the following sum converges: $\sum_{t=1}^{\infty}{\left(\frac{1}{2}\right)^t} \ln\left(\frac{W+2^{t-1}-P}{W}\right)$ The author writes: "This sum converges (as long as each individual term…
Julian Rodemann
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Is this convergent sum a constant?

I got this symbolic convergent sum from $\textit{Mathematica}$: $$\sum _{k=1}^{\infty } \frac{k!}{(2 k)!}=\frac{1}{2} \sqrt[4]{e} \sqrt{\pi } \text{erf}\left(\frac{1}{2}\right)$$ Where $\text{erf}\left(\frac{1}{2}\right)$ can be found here. Is this…
Fred Daniel Kline
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Set convergence

I was wondering how to correctly mathematically describe the following observation/fact. Let us consider the set of points on the real line defined as $\frac{K}{n}$, where $K$ is a chosen real constant and $n \in \mathbb{N}$. Let us take $K=1$: the…
An aedonist
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Prove that the given sequence converges

Prove that the given sequence $\{a_n\}$ converges: $a_1 > 0, a_2 > 0$ $a_{n+1} = \frac{2}{a_n + a_{n-1}}$ for $n \geq 2$ As I observed, this sequence does not seem to be monotonic and that it could be bounded since the values of $a_1$ and $a_2$ are…
Aid
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Investigate the convergence of $ \sum_{n=1}^{\infty } (-1)^{n}\frac{n+2}{n(n+1)} $

I am supposed to investigate the convergence of $ \sum_{n=1}^{\infty } (-1)^{n}\frac{n+2}{n(n+1)} $. I'm unsure whether to use Leibniz' criterion or a comparison test and I really can't start. Thanks
J. Doe
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Prove that$\sum_{n=1}^{\infty} 2^{-2^{n}}$ converges to an irrational limit.

Prove that$\sum_{n=1}^{\infty} 2^{-2^{n}}$ converges to an irrational limit.
You Zhou
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Clarification on absolute / conditional convergence test

I understand that given the absolute convergence test, if I am able to prove that the absolute of the series converges, then the series itself will converge itself as well. What if I want to prove for conditional convergence? Is it sufficient to…
statsguy21
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For how many values of b mod 55 does the congruence x^2 + x + b = 0 (mod 55) have exactly 2 solutions.

Question: For how many values of b mod 55 does the congruence x^2 + x + b = 0 (mod 55) have exactly 2 solutions? I tried to use quadratic function to solve but really don't get it.
Zhenqing Xu
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Mean square convergence of Parzen Windows

I am studying the Parzen Windows technique. Although I could find and understand an outline of the proof (for the convergence in mean square), e.g. that provided in Duda's book or sketched at…
Julien
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Convergence test of $\sum_{n=2}^{\infty} \frac {1}{\ln^n(\frac{1}{n})} $

I need to test for convergence and absolute convergence this sum: $\sum_{n=2}^{\infty} \frac {1}{\ln^n(\frac{1}{n})} $. Thank you for your help.
Tomáš Svoboda
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Prove the following problem about fibonacci convergence problem

Determine whether the series is convergent or divergent. Give reasons for your answers. $$\sum_{n=1}^\infty \frac{1}{a_n}$$ where $a_1=a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for all $n\ge1$. umm... How do I solve this? I don't even know how to start...
Applepie
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Cauchy comparison test - if the series limit tends to 1 from bottom

I've a following task Check convergence of: $$\sum_{n=2}^\infty (-1)^{\lceil 1+\sin^{2} n^{5} \rceil}\left(\frac{n^{2}+3n+10}{n^{2}+5n+17}\right)^{n^{2}(\sqrt{n+1}-\sqrt{n-1})}$$ My solution is: $\lceil 1+\sin^{2} n^{5} \rceil$ is always 2, because…
Joggi
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Multiplication of matrices ${\bf W}$ and ${\bf W}^H$, which are the span of nullspace

When ${\bf W}\in \mathcal{C}^{m\times (m-n)}$ is a nullspace matrix of ${\bf A}\in\mathcal{C}^{n\times m}$, where $m>n$, I have confirmed that ${\bf W}^H{\bf W}={\bf I}_{m-n}$. Here, ${\bf I}_{m-n}$ is an ($m-n$)-dimensional identity matrix. Now, I…
Math
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About convergence of logs

I have a function $f(c,x)=\log(c^x/x)$ I want to know the limit of this function when $c$ is constant, but $x \to \infty$. Simulations suggest it tends to infinity, but I'd like to have a formal argument.
fox
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Please do these series diverge or converge and find their sums if they converge

a) $$\sum_{k=1}^\infty 3^{-2k+1}$$ b) $$\sum_{k=1}^\infty 3^{2k+1}$$ my Trial a) $$\sum_{k=1}^\infty 3^{-2k+1} = \sum_{k=1}^\infty \frac{-3}{9^k}$$ I am blocked because I wanted to have it in the form $ar^n$ and later us
Alio Badi
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