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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 .

21990 questions
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Known convergence of 2 series, prove bijection

Here is a question that confused me much. Actually I can see the bijection, but don't know how to prove it. Here is a theorem that might useful, http://amininima.wordpress.com/2013/05/08/conditional-convergence/ but it uses bijection as a known…
asaak
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Find convergence of $\sum_{n=1}^{\infty}\frac{1}{n^2-ln(n)}$

$$\sum_{n=1}^{\infty}\frac{1}{n^2-ln(n)}$$ I was comparing it to $$\sum_{n=1}^{\infty}\frac{1}{n^2}$$ and I had limit $$\lim_{n\to \infty}\frac{a_n}{b_n}$$ where $${a_n}=\frac{1}{n^2-ln(n)}$$ and $${b_n}=\frac{1}{n^2}$$ but at the end I have that…
Akane123
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Convergence Test for Series $\displaystyle\sum_{n=1}^{\infty} \frac{n^{n}}{(n+1)^{n+1}}$

I am trying to determine the convergence of the series: $$ \sum_{n=1}^{\infty} \frac{n^{n}}{(n+1)^{n+1}} = 1 + \frac{1^1}{2^2} + \frac{2^2}{3^3} + \frac{3^3}{4^4} + \frac{4^4}{5^5} + \dots \text{(to infinity)} $$ I've attempted to apply D'Alembert's…
Shreyaskar Saha
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How do we argue that in math-terms?

If we have $\,\lim\limits_{n\to \infty}\dfrac{\log n}{n}\,,\,$ I know that this is converging to $0$. I know that $\,n\,$ is a "stronger function" than $\,\log n\,,\,$ but is there a "mathematical way" of saying that ?
kasper kaspersen
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proving or disproving conditional convergence

Prove or disprove that if $\sum \frac{n a_n}{2^{n-1}}$ diverges then $\sum (-1)^n a_n$ could be conditionally convergent. Where to start? I don't see the connection between those two series. btw, how do I write An like in recursive series? didn't…
Ori Yehuda
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Statistic convergence as surely question

Suppose $X_n$ converges almost surely to $X$, and $f$ is a continuous function. Prove that $f(X_n)$ converges almost surely to $f(X)$. My approach: according to definition of continuous $|x - a| < \alpha$, then $|f(x) - f(a)| < \delta$ so that take…
Smith
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Statistics: convergence in distribution

A roulette wheel has 18 red wedges, 18 black wedges and 2 green wedges. Wedges are equally likely to come up in one spin of the wheel. A gambler bets one dollar on red coming up in independent spins of the wheel. Briefly explain why the gambler…
Smith
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Given $\sum^{\infty}_{n=1}\frac{n^{2}+(-1)^{n}}{2n^{3}+1}$, is this absolute convergent?

I already know that the sequence converges as when $n$ approaches $\infty$: $$\lim_{n \to\infty}\frac{\frac{n^{2}}{n^{3}}+\frac{(-1)^{n}}{n^{3}}}{\frac{2n^{3}}{n^{3}}+\frac{1}{n^{3}}} \;\;\implies\;\;\lim_{n \to…
Sleepy
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Show that the sequence {$a_n$} converges to $a \in \mathbb {R}$ $\leftrightarrow$ all real subsequence {$a_{n_k}$} of {$a_n$} converge to $a$

I need help with the following two tasks: a) Show that the sequence {$a_n$] converges to a $\in \mathbb {R}$ $\leftrightarrow$ all proper subsequence {$a_{n_k}$} of {$a_n$} converge to a. Well the right direction $\rightarrow$ is easy to proof. If…
Analysis_Mark
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Convergence radius of a power series $\sum_{k=0}^\infty a_kx^{2k}$ with $\lim\limits_{n \rightarrow \infty}|a_n|^{\frac{1}{n}}=\frac{1}{r}$

I need help for the following task: Let {$a_n$}$_{n\in\mathbb{N}}$ be a sequence of real numbers such that $\lim\limits_{n \rightarrow \infty}|a_n|^{\frac{1}{n}}=\frac{1}{r}$ with $r$ being a positive real number. Show that the power series…
Analysis_Mark
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When does this integral converge $\int_{0^+}^{1}{\frac{x^a\ln(x^b)}{e^x-1}}dx$?

$$ \int_{0^+}^{1}{\frac{x^a\ln(x^b)}{e^x-1}}dx,\quad \forall a,b \in \mathbb{R},a\ge 1 $$ It's from a final exam. I just can't find a proper function to prove convergence or divergence. For $b=0 \rightarrow \frac{x^a\ln(x^b)}{e^x-1}=0 \therefore$…
Pedro Giuttari
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How do I prove this statement about convergence in L^1?

I have the following problem: Suppose that $f,f_1,...,f_n,...\in L^1(\Omega)$ and $f_n\stackrel{a.e.}{\rightarrow}f$. Show that $$f_n\stackrel{L^1}{\rightarrow}f\Leftrightarrow ||f_n||_1\rightarrow ||f||_1$$If possible use the functions…
user123234
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Convergence - formula

There are sequences: $\{x^n\}_{n\in N}$, where $x^n=\langle x^n_1, x^n_2, x^n_3,...\rangle, n=1,2,...$ $x=\langle x_1,x_2,x_3,...\rangle$ How should I write that $x$ is limes of $x^n$? I use definition that $lim_{n \rightarrow \infty}x_n=x$ iff…
user23709
  • 759
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limes inferior & limes superior

If $U=[-1,1]$ and $J(u)=u, u\in (0,1]$ $J(u)=a, u=0$ $J(u)=1-u, u\in [-1,0)$ how to calculate limes inferior and limes superior of $J(u)$? Is this correct: I choose arbitrary sequence $\{u_k\}\in U$ such that $\displaystyle \lim_{k \to…
user23709
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Convergence of double sum on the lattice

I am working with a commutator $T$ acting on the lattice $\ell^2(\mathbb{Z}^2;\mathbb{C})$, the function space made up by the basis elements \begin{align}\left|\vec{x}\right>\,:\,\mathbb{Z}^2&\rightarrow \mathbb{C}\\ \vec{y}&\mapsto…
mk_math
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