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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 .

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Convergence of $\sum_{n=0}^\infty \frac{1}{n^n}$

I'm stuck at deciding wether or not $\sum_{n=0}^\infty \frac{1}{n^n}$ converges.The sequence itself is a zero sequence and the root test seems to pass, but how can that be since for n=0 we would have have to deal with point of singularity.
smihael
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convergence $\sum^{\infty}_{k=1}\frac{k^2+3k+1}{k^3-2k-1}$

Finding whether the series $$\sum^{\infty}_{k=1}\frac{k^2+3k+1}{k^3-2k-1}$$ is converges or diverges. What i try $$\frac{k^2+3k+1}{k^3-2k-1}\approx\frac{k^2}{k^3}=\frac{1}{k}$$ So our series seems to ne diverges. But i did not understand How do i…
jacky
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Proof of limits of sequences tending to infinity

How can we prove this : Prove that if $\quad\lim_{n\to+\infty} y_n = \lim_{n\to+\infty} z_n=+\infty$ then $\quad\lim_{n\to+\infty} v_n = \lim_{x\to+\infty} w_n=+\infty$ . With $$w_n=\frac{n}{\sum_{i=1}^n\frac{1}{z_i}},\quad v_n =…
14max
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radius of convergence of series having factorial

Finding radius of convergence of following series $$f(x)=\sum^{\infty}_{n=0}\frac{4^n\cdot n!}{(2n+1)}x^{2n+1}$$ What i try : let $\displaystyle a_{n}=\frac{4^n\cdot n!}{(2n+1)}x^{2n+1}$. Then $\displaystyle a_{n+1}=\frac{4^{n+1}\cdot…
jacky
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convergence or divergence of logarithmic sum

Finding whether the series $$\sum^{\infty}_{n=1}\frac{2}{n(1+\ln(n))^{3}}$$ is converges or Diverges. I am trying to solve it using Integral test Let $\displaystyle f(x)=\frac{2}{x(1+\ln(x))^3}=\frac{(1+\ln(x))^{-3}}{x}.$ Then $\displaystyle…
jacky
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Value of $\sum\limits_{n=0}^{\infty}(n+1)(0.8x)^n$

I know $\sum\limits_{n=0}^{\infty}(n+1)x^n = \frac{1}{(1-x)^2}$ but does changing to $\sum\limits_{n=0}^{\infty}(n+1)(0.8x)^n$ make a difference? wolfram seems to be giving me a much more complicated result.
user630591
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An elementary question on radius of convergence.

Is the radius of convergence of $\sum_{n=0}^{\infty} {z^{n^4}} $ = 1? By definition it is the reciprocal of nth root of supremum of coeff of $z^n$, so should be 1 here. Am I correct?
user67773
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Order of convergence in a linear equation

This is a general question about using the $\mathcal{O}$ idea to say something about the sign of coefficient. Suppose I have $$x:=\frac{1}{b^2}\gamma(b)-\frac{1}{b^3}\beta(b)$$ and $b\in (b_0,+\infty)$ where $b_0 >1$. $\gamma(b)$ and $\beta(b)$ are…
kangkan Dc
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How to show that $f_n(x)=1-x^n$ converges to a set valued function, $f(x)$, which takes 1 for $x\in[0,1)$ but takes $[0,1]$ at $x=1$?

It seems that $f_n(x)=1-x^n$ converges to $f(x)$ such that \begin{eqnarray*} f(x)&=&1~~\text{for}~~x\in[0,1)\\ &=& [0,1]~~\text{for}~~x=1.\end{eqnarray*} I wonder if it is correct or not. If it is, how to show this convergence?
user722137
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The summation of reciprocal of factorial of number

$$\lim_{n\to \infty} S_n=\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{n!}$$ I need to show that this series is a convergent series. How do I show this series to be convergent? My book says that this is convergent. Please provide sone hint.
The Learner
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convergence: conditional and absolute

Some sources say that the limit to infinity needs to be 0 to someone qualify for 'convergent'. Some source say that it only needs to go to one number. Could someone explain this?
user68610
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Is $(2n)!$ the same as $2(n!)$?

I am trying to determine the convergence of a series $$\sum_{n=17}^{\infty} \frac{(n!)}{(2n)!}.$$ Using the ratio test, I have simplified $a_{n+1}/a_n$ to $$\frac{(2n!)}{2(n)!}.$$ If $(2n)!$ is the same as $2(n!)$, I can strip out the factorial to…
go-already
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Convergence of sequence of random variable

Let $(U_n: n\geq 1)$ be a sequence of independent random variable and each distributed uniformly on the interval $[0\;1]$. Let $X_0=0,$ and define $X_n$ for $n\geq 1$ by the following recursion: $X_n=\max\left\{X_{n-1}, \frac{X_{n-1}+U_n}{2}\right…
Suyog Kapsikar
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Convergence of product of rational functions

How can I prove that the sequence $a_{n} = \frac{4}{5} * \frac{104}{105} * ... * \frac{50n^{2} - 50n + 4}{50n^{2} - 50n + 5}$ converges or not (where $a_{1} = \frac{4}{5}$, $a_{2} = \frac{4}{5} * \frac{104}{105}$, and $a_{3}= \frac{4}{5} *…
324
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Convergence of locally integrable functions

Given that $f_j, g_j$ and $f,g$ are locally integrable functions, i.e. they are in $L_{loc}^1(\mathbb{R})$. Under the assumption that $f'_j(x)=g_j(x)$ in the sense of distributions, I wanna show that $f'(x)=g(x)$ in $\mathbb{R}$ in the sense of…
Jonathan Kiersch
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