# Questions tagged [absolute-convergence]

This tag is for questions related to absolute convergence of a series.

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### What can be said about the sum of the series?

Let $\{a_n \}_{n \geq 1}$ be a sequence of non-zero integers satisfying I. $|a_n| \lt |a_{n+1}|,$ for all $n \geq 1$ II. $a_n$ divides $a_{n+1},$ for all $n \geq 1$ and III. every integer is a ...
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### Show that $\sum\frac{(-1)^{n+1}} {{n}^r} \sum\frac{(-1)^{n+1}} {{n}^s}$ by Abel's rule forms a series that doesn't converge when r+s=1.

It is a similar problem to that in Show that the series $\frac{1} {\sqrt{1}} -\frac{1} {\sqrt{2}} +\frac{1} {\sqrt{3}} +\dots$ converges, and its square (formed by Abel's rule) doesn't.. It ...
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### Prove: if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely

Prove: if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely. I get why the statement is correct (because it means that the convergence of the series ...
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### Show that the series $\sum\limits_{n\ge1}a_nX_n$ converges absolutely a.s. for some constants $a_n\ne0$.

Let$\{X_n\}_{n\ge1}$ be an arbitrary sequence of random variables. Show that the series $\sum\limits_{n\ge1}a_nX_n$ converges absolutely a.s. for some constants $a_n\ne0$. My attempt at a solution: ...
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### Convergence of double sum and its rearrangement

Let $A=\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij},a_{ij}\geq 0$ and $$\phi:\mathbb N \to\mathbb N\times\mathbb N$$ be any bijection. Now $B=\sum_{k=1}^{\infty}a_{\phi(k)}$. How $A$ and $B$ are ...
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### For which $x$ is $\sum_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$ (absolutely) convergent?

Consider the power series: $$\sum_{n=1}^{\infty} \frac{(x-b)^{n}}{na^{n}}$$ with a, b >0 : a) for which x is this series absolute convergent, b) for which x is this series conditionally convergent, ...
Boiled down, I have two functions of x and I need to know if the indefinite integral of their product converges. In particular, if: $$y=x\exp{-x^2}$$ then how can I show (if its true) that the ...