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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question about applying a comparison test to a sequence

In lecture, we were asked: Does $ \sum \limits_{n=0}^\infty \frac{ 2n^3+ 3n -8 }{ n^5-5n^3-n^2+2 }$ converge? $a_n = \frac{ 2n^3+ 3n -8 }{ n^5-5n^3-n^2+2 }$ In discussing strategies for applying a comparison test for $a_n$ we rejected several…
maogenc
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Convergence of generalized sums of non-zero elements

I believe the following little theorem is valid, but the premises are ugly, and I'd like some advice on improving them: Theorem: Let $G$ be a totally ordered, Archimedean, and abelian group with identity $0_G$, which is first-countable when…
dfeuer
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Prove that $\{x_n\}_n$ is convergent.

Here is an exercise: Let $x_n=1+2+\dots+\frac1n-\ln n$. Prove that $\{x_n\}_n$ is convergent. (I believe that this can be found in the site, however I cannot find immediately, so I post it here.) The hints are much appreciated. I don't want…
Paul
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does $ \sum _{n=0}^{\infty }\left(\cos^n\left(\frac{1}{\sqrt{n}}\right)-\frac{1}{\sqrt{e}}\right) $ converge?

I'm trying to find out whether $\sum _{n=0}^{\infty }\left(\cos^n\left(\frac{1}{\sqrt{n}}\right)-\frac{1}{\sqrt{e}}\right)$ converges or not. I've tried with taylor series but it doesn't lead me anywhere except with the fact that $\lim_{n \to…
sky1099
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Convergence of stochastic process

I'm stucked with this exercise. Hope someone can help me $X_t = 3+a_t$ with $a_t \sim (0,\sigma^2)$. $a_t$ is iid. $Y_n=\frac{1}{\sqrt{n+2}} \sum_{t=1}^{n} X_t $ Does $Y_n$ converge in distribution? What distribution?
sgarg96
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Convergence by using Cauchy Criterion

this is the sequence: $(a_n)=\frac{1}{n+1}+\frac{1}{n+2}+\cdot\cdot\cdot+\frac{1}{2n}$ And this is what I tried to do so far: $|a_{n+1} - a_{n} | = \frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1} = \frac{1}{2n+1}-\frac{1}{2n+2}$ in order to show…
Vazrael
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Does the generalised integral $\int_{0}^{\infty}\frac{e^{\arctan(x)}-1}{x \sqrt x}dx$ converge or diverge?

Does the generalised integral $$\int_{0}^{\infty}\frac{e^{\arctan(x)}-1}{x \sqrt x}\,dx$$ converge or diverge? The first thing I do is divide it into two integrals $\int_{0}^{A}\frac{e^{\arctan(x)}-1}{x \sqrt x}\,dx$ +…
PythonDaniel
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Does the series $\sum_{n=1}^\infty \sqrt {1- \cos(\pi /n)}$ converge or diverge?

So I want to decide if this series converges or diverges $\sum_{n=1}^\infty \sqrt {1- \cos(\pi /n)}$. My initial thought is that I should calculate $\lim_{n\to\infty} \sqrt {1- \cos(\pi /n)}$ which approaches zero because $\sqrt{1-1=0}$ and then it…
PythonDaniel
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Show that the sequence doesn't converges under discrete metric

I have the sequence $ x_n:=\left(1-n^{-n}, 2-e^{-n}, 3-2^{-n}\right) $ in $ \mathbb{R}^3 $ and consider the discrete metric $ d(v,w)=\begin{cases}0,\quad v=w\\1,\quad v\neq w\end{cases} $. I want to show that this sequence does not converges to $…
hallo007
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Proof of convergence of $\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor n\sqrt{2}\rfloor}}{n}$

I had this task long long time ago in a calculus class. I remember it had a very elegant solution using nice property of $\text{Spec}(\sqrt{2})$, where $$ \text{Spec}(\alpha)=\{\lfloor\alpha n\rfloor\,:\,n=1,2,3,\ldots\}. $$ I wasn't able to…
kubus
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Multiple choice question - series convergence

Say we have the series $a_{n}>0$. Let's define $b_{n}=\int_{0}^{a_{n}+1}x^{n-1}dx$. Therefore: a. $\liminf(nb_{n})\geq1$. b. If $b_{n}$ converges then $\limsup na_{n}\leq 1$ c. If $\liminf a_{n}\ > 0$ then $\lim b_{n}=\infty$ d. If $\limsup b_{n}<…
Math101
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convergence or divergence of Irrational series.

Finding convergence or divergencs of series $$\sum^{\infty}_{n=1}\sqrt{\frac{4n^6+3n}{2n^2+n+5}}$$ What i try:: $$\frac{4n^6+3n}{2n^2+n+5}\approx 2n^4$$ $$\sqrt{\frac{4n^6+3n}{2n^2+n+5}}\approx \sqrt{2}\;…
jacky
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given a divergent series, can we conclude a related sequence is not converging to zero?

say we have a sequence of non-negative reals, $a_1, a_2, \dots$, and that $\displaystyle\sum\limits_{n=1}^{\infty}a_n$ is divergent, meaning convergent to infinity. Under this scenario I am trying to prove that the following sequence in $m$ cannot…
311411
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Comparison test for sin

I need to use the comparison test for convergence on $\sum_{n=1}^\infty 2^n\sin\frac{(a)}{3^n}.$ I have no idea how to tackle this. Some help/hints would be greatly appreciated. Thank you!
Stefana
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Prove that $\frac{1}{x_n}\to\frac{1}{2}$ if $x_n\to 2$.

No limit theorems allowed. Thanks! We know it is something like $$\left|\frac{1}{x_n}-\frac{1}{2}\right|=\left|\frac{2-x_n}{2x_n}\right|<\epsilon$$
Emily
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