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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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sequence of monotone functions on a compact interval converges uniformly to a *monotone* function?

Let $(f_n)$ be a sequence of monotone functions on a compact interval $I$ which converges pointwise to a continuous function $f$. Show that $f$ is monotone and that $(f_n)$ converges uniformly to $f$. I am unable to show that $f$ is monotone,…
macton
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How to determince convergence and divergence of

How to determine convergence and divergence of $a_n =\sqrt[\Large^n]c$ ?
omidh
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For what a will this series converge?

$\sum_{n=1}^{\infty}n^a\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ I tried to use the D'Alembert ratio test and it didn't quite work out. The series inside is telescoping but I don't know if that information will be useful. Can someone…
marcopopo
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Finding the lowest value for $a$ at which $\lim_{n\to\infty}\frac{(2n)!}{(n!)^a}$ is convergent

So I've noticed that for larger values of $a$ the function below forms a bell curve shape whereas for lower values it diverges, I would like to find an approximation for the lowest value at which this is true: $$f_a(x)=\frac{(2x)!}{(x!)^a}$$ I have…
Henry Lee
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$\sum_{k=2}^\infty \frac{1}{1+(-1)^k \sqrt k}$ converges or diverges?

$\frac{1}{1+\sqrt k (-1)^k}= \frac{1-(-1)^k \sqrt k}{1-k}$. Not quite sure how to move on from here.
Labbsserts
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Cauchy multiplaction

Would like to know if exists an example for $$\sum_0^\infty a_n x^n,\sum_0^\infty b_n x^n$$ $$\sum_0^\infty c_n x^n, c_n:=\sum_{k=0}^n a_k b_{n-k} $$ such that $\max\{R_a,R_b\} < R_c < \infty$ ($R$ stands for convergence radius) ?? (Couldn't find…
ORBOT Inc.
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What is the range of $a $ such that $\frac{\log N}{N^{3a+1/2}}\rightarrow 0$ as $N\rightarrow \infty$?

I want to find the range of $a$ such that $\frac{\log N}{N^{3a+1/2}}\rightarrow 0$ as $N\rightarrow\infty$. The answer to this question hinges on conditions under which $logN$ diverges slower than the numerator, which is a power function of $N$. Is…
ExcitedSnail
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Determine if the series $\frac{((\ln(n))^3}{n}$ is convergent or divergent.

Determine if the series $\frac{((\ln(n))^3}{n}$ is convergent or divergent. What I Tried :- If I were to use the comparison test would I end up with $(\ln(n))^3 > 1/n^2 > 0$. So $\frac{1}{n^2}$ is convergent by $p$-test as $(p=2>1)$. Therefore the…
MomLearningMath
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Is the sequence always divergent

Consider the sequence $x_n = [(\frac{2}{n}+1)^r+(\frac{2}{n}-1)^r]n^{r-1}$, $n=1,2,3,..., r=1,2,3...$ Is it always true that $x_n \rightarrow\infty$ as $n\rightarrow\infty$? Or is it true only when $r$ is even number and does it converge to $0$ when…
user587389
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Convergence of $\sum_{k=1}^n\frac{(-1)^k}{\sqrt{k}}$

Does $\sum_{k=1}^n\frac{(-1)^k}{\sqrt{k}}$ converge? My attempt: $$\begin{aligned} \sum_{k=1}^{2n}\frac{(-1)^k}{\sqrt{k}}&= \sum_{k=1, 3, ..., 2n-1}\left(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\right)\\ &=\sum_{k=1, 3,...,…
Tomato
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Sequence of Gaussian random variables

Let $(X_n)$ be a sequence of Gaussian random variables $N(m_n, \sigma_n)$. Let's say there is a random variable X sucht that $X_n$ goes to X in distribution ($n \rightarrow \infty$). In this case, we have: $\mathbb E(e^{itX_n}) \rightarrow \mathbb…
JohnD
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Examining convergence of the series.

Examine convergence of the series $$1-\frac{1}{2}+\frac{1\cdot3}{2\cdot4}-\frac{1\cdot3\cdot5}{2\cdot4\cdot6}+\cdots$$
Soham Sarkar
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values of q for which tangent integral is converges

Finding value of $q$ for which $$\int^{1}_{0}\frac{1}{(\tan (x))^{q}}dx$$ converges What i try:: Let $\tan x=t.$ Then $\displaystyle dx=\frac{1}{\sec^2 (x)}dx=\frac{1}{1+t^2}dt$ And changing limits $$I=\int^{\tan…
jacky
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Integral test fails?

I have a problem with a series. We have the series $$\sum \limits_{n=1}^{+\infty} \frac{1}{(n+2)(n+4)}.$$ When I used the integral test for convergence (correct me if I'm wrong), the result is $$\lim_{b \to +\infty} \left[ \frac{1}{2} \ln…
Lourd John Joaquin
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convergence of $ \sum_{n=1}^{\infty} \frac{n !}{n !+3} $?

determine the convergence of $$ \sum_{n=1}^{\infty} \frac{n !}{n !+3} $$ I tried using the ratio test and also for n! , I use Stirling approximation.Still I got stuck.
user791345
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