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
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Ratio Test and L = 1

Consider the series $$\sum\limits_{n=1}^\infty\frac{(2n)!}{a^n(n!)^2}$$ with $a > 0$. Determine if the series converges for: i) a > 4 ii) 0 < a < 4 iii) a = 4 For i) and ii), I will use the ratio test,…
darkchampionz
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Is every infinite decimal sequence convergent?

Is every infinite decimal sequence convergent? For example, would this sequence $x= 0.12112211122211112222\ldots$ be considered convergent?
user833684
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Proof of convergence in distribution of a discrete random variable

I'm working on a question on "convergence in distribution" and I appreciate if you could guide me on how to approach this question: Here is the question: Let $X_n$ be integer-valued random variables. Show that $X_n \stackrel w{\longrightarrow}…
user48405
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pointwise convergent subsequences of $f_n(x)=(-x)^n, x\in [0,1]$

I want to find its pointwise convergent sequence, if any. $f_n$ itself converges pointwise everywhere, and there will be a pointwise convergent sequence of odd and even, so at least two.
blabla
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Radius of Convergence without Ratio Test

I've reached my wit's end with this problem. We have not been taught about Ratio Tests or anything involving Radius of Convergence, but we have a problem asking for it. For what values of $p$ does the sum converge? $$\sum_{n=1}^\infty{\ln(n)…
user319742
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Counter-example for misused theorem in series convergence test

I had a math exam not so long ago and got my result back, I'm happy with the result but there is a question for which my teacher gave me an explanation (for me not having the points) but I still think my reasoning is good. So I am asking for a…
Sebastien Comtois
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How to prove the convergence of $\sum_{n=1}^\infty (-1)^n\frac{\ln n}{n}$?

$$\sum_{n=1}^\infty (-1)^n\frac{\ln n}{n}$$ The above series is the one I want to prove convergent. I took $\frac{\ln n}{n}$ but I didn't find anything that I could do with it. I tried to compare it with another series, didn't find a good one. Any…
Andrew
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Prove the convergence of : $\sum \ln(n)/n^{3/2}$

I've been having some issues with what to compare it to. I have a hunch it converges. But I just cannot figure out what I can compare it too. Please help. :)
E.C.
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Given that $X_i$ are symmetric about 0 and iid with $E[|X_i|]=\infty$. Show that $\frac{S_n}{n}=\frac{X_1+\dots+X_n}{n}$ does not converge to 0.

I've been trying something like.. Let us assume that $\frac{S_n}{n}$ converges to 0. That means that, $$\frac{S_{n+1}}{n+1}-\frac{S_n}{n}$$ converges to 0 too. But we can rewrite this as…
user132883
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Prove that $x^n/n!$ converges to $0$ for all $x$

Prove that $a_n=x^n/n! \to 0$ for all $x$ Here is what I tried, but it seems to lead to nowhere. Choose $\epsilon > 0$. We need to show that there exists $N\in \mathbb{N}$ such that for all $n>N$ we have $|a_n| < \epsilon$ So, $|(x^n/n!)| <…
user133993
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Counterexample of bounded functions

${f_n}$ is a sequence of continuous functions on R, and $f_n$ converges to f uniformly on R. If each of the functions $f_n$ is bounded, show that this does not imply that f is bounded. Please help with the counterexample - I don't understand how to…
kiwifruit
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Uniformly convergent sequence

$\{f_n\}$ is a sequence of continuous functions such that $f_n \to f$ uniformly on $\mathbb{R}$. Suppose that $x_n \to x_0$, prove that $\displaystyle \lim_{n \to \infty} f_n (x_n) = f(x_0)$. I think I must be missing something here, but I don't see…
kiwifruit
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On strong convergence

Someone can I help about the question: Let $E=C[0,1]$ be the space of continous functions defined on unital interval. Consider the sequence $\{f_n\}$ in $E'$ by $\langle f_n,\varphi \rangle = n \int_{0}^{\frac{1}{n}}\varphi(t) dt$ with $\varphi \in…
Silvio Sandro
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Convergence or divergence of the sum series with $\sum_{n\ge 1} \frac{(1-2^{1/n})^6}{(n+3)^p}$, $p\in \Bbb R$

I am stuck on finding the convergence or divergence of $$\sum_{n=1}^{\infty} \frac{(1-2^{1/n})^6}{(n+3)^p}$$ depending on p with p being in R. I tried to compare it and say its general term is smaller than $$\frac{1}{{(n+3)^p}}$$ and then consider…
Haxel
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For what value of C does this pattern converge?

If we have a series of terms that follows the pattern of $x_{i}=\frac{x_{i-1}+x_{i-2}}{Cx_{i-1}},$ the pattern eventually diverges towards two oscillating values, for most values of $C$. Given that this series should converge towards $2/C$ as $i$…
Take
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