# Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

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### Prove that if there exists a subsequence of $a_{n}$ which converges to $L$ , then $L$ is a limit point of $a_{n}$.

Let $(a_{n})_{n=0}^{\infty}$ be a sequence of real numbers, and let $L$ be a real number. Then the following statements are logically equivalent: (a) $L$ is a limit point of $(a_{n})_{n=0}^{\infty}$. (...
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### Showing a series converges absolutely

The goal is to prove that if $|\frac{c_{n+1}}{c_n}|\leq1+\frac{a}{n}$, where $a<-1$ and $a$ does not depend on $n$, then the series $\sum_{n=1}^\infty c_n$ converges absolutely. My idea: to have ...
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### What is the theory of the cones of minorant functions?

As a non - mathematician, I face some difficulty in understanding concepts such as convex cone, minorant/majorant function. The question is What is the theory of the cones of minorant functions? And ...
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### How can i find the sum of the infinite series? [duplicate]

I am having trouble finding the sum of this series. I was able to show that it is convergent by Ratio Test but i could not find the sum of this series. I would appreciate it very much if you could ...
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### Convergence of an improper triple integral

Let $$V=\{(x,y,z) \in \mathbb{R^3} : x\ge 1,y>0, x^2+z^2<\mathrm{min}(2x,y^2)\}$$ be a region of the real space. For which values of $\alpha \in \mathbb{R},$ $\iiint_V |z|y^{-\alpha}$ converges (...
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### Series convergence test, $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$

I'm trying to find all $x$ for which $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$ converges. I know I need to check the ends ($-1$ and $5$) but I'm not sure what to happen after that. I'm pretty sure I'...
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### What's this partial sum $\sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$ equal?

I want to get this partial sum of $$\sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$$ which it is convergent and it is closed to one half , I have tried to use polylogarithm function which is defined as :...
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### Question on a problem regarding improper integral

I'm studying improper integrals with Paul's Online Notes as a reference. Sorry if I'm quoting it here, but the website has the following problem: Determine if the following integral is convergent or ...
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### Is $\sum_{n=1}^{\infty}$ $(\sqrt[\leftroot{-3}\uproot{3}n]{a}-1)^p$ divergent or convergent?

I know $a>1,p>0$. I've tried the direct comparison test, but I can´t find a sequence $b_{n} > a_{n}$ whose sum converges, or a sequence $b_{n} < a_{n}$ whose sum diverges.
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### Probability convergence and almost surely convergence [closed]

Let $(\Omega, \Im ,\mathbb{P})$ be a probability space and let $\left\lbrace X_{n} \right\rbrace {}_{n}$ be a sequence of random variables such that $$\mathbb{P} \left(X1>X2>...>0\right)=1$$ ...
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### Lipschitz Constant for Equation System

I'm would like to compute the region of convergence from Newton-Rapson Method for a multivariate system of equations by a theorem that states: Let $\vec{F}: \mathbb{R}^N \rightarrow \mathbb{R}^N$ be ...
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### Abbott's proof that any rearrangement of an absolutely convergent series converges to the same limit as the original

Here is his proof in full: Assume $\sum\limits_{k = 1}^{\infty} a_k$ converges absolutely to $A$, and let $\sum\limits_{k = 1}^{\infty} b_k$ be a rearrangement of $\sum\limits_{k = 1}^{\infty} a_k$. ...
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### Question regarding the correlation of limit points and convergent sequences

I've been told, that: $(i)$ $M$ is a closed set $(ii)$ For every convergent sequence $a_{n}$ in $M$, $\lim \limits_{n \to \infty } a_{n} \in M$ are equivalent. I understand that $M$ being a closed ...
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The given series: $$\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \cot^n x}$$ Here is what I did: $$\sum^{\infty}_{n=1} \frac{\cos (\pi n) \sin \left(\pi x \right)}{(n+1)n \... 1answer 70 views ### Whether \lim_{n\to \infty} \frac{2}{\mathsf{e}}(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k}(1-\frac{2k}{n})^{n-1})^{-1/n} exists Problem: Decide whether or not \lim_{n\to \infty} \frac{2}{\mathsf{e}}\left(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k} \left(1-\frac{2k}{n}\right)^{n-1}\right)^{-1/n} exists. Background ... 1answer 25 views ### Convergence Theorem for Power series : radius of convergence and normal convergence I am currently working through the textbook "Complex Analysis" by Freitag and Busam. Proposition III.2.1 (Convergence Theorem for Power Series) reads : For each power series$$ a_{0} + a_{1} ...
For $n \in \mathbb N$ $f_n (x) = \frac{n^2 }{(x^2 +n^2)}$ Let $x \in \mathbb R$. Prove that $f_n (x) \rightarrow 1$ as $n \rightarrow \infty$ My attempt: Let $\epsilon>0$, then by definition of ...
Does the following integral converge for $x < 0$ $$\int _x^0\:\cfrac{\ln^2 \ (|t |)}{ \sqrt{t} }dt$$ I tried splitting it into two separate integrals under the assumption that both are ...