# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

421 questions
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### Does an unconditionally convergent series of complex numbers converge absolutely?

Suppose $\sum a_n$ is a series of complex numbers, if $\sum a_n$ and its every rearrangement all converge to the same sum, does $\sum a_n$ converge absolutely?
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### prove/disprove absolute convergent

$\int_0^\infty 3^{-x}x^4cos(2x)dx$ I succeeded to prove that this integral is conditionally convergent with Dirichlet's test. I don't know how to prove/disprove absolutely convergent.. Thanks !
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### Show that the interval of convergence for $\Sigma\frac{\sin(nx)}{n^2}$ is $\mathbb{R}$

Let $f_n(x) = \frac{\sin(nx)}{n^2}$. I want to show that the infinite series $\sum f_n$ converges for all $x$. After trying the ratio test and getting nowhere, I attempted to use the comparison test ...
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### Prob. 9, Chap. 6, in Baby Rudin: Which one of these two improper integrals converges absolutely and which one does not?

Here are the links to three earlier posts of mine on Prob. 9, Chap. 6, in Baby Rudin here on Math SE. Prob. 9, Chap. 6, in Baby Rudin: Integration by parts for improper integrals Prob. 9, Chap. 6, ...
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### Series of Cauchy Products

I am trying to figure out convergence conditions for the series $B\equiv \sum_{n=0}^\infty A_n A_n'$ where $A_n = \sum_{k=0}^\infty a_{n+k}$ and $A_n'$ being the transpose of $A_n$. I have a number ...
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### In the alternating series test, what is the condition for convergence? [closed]

When testing for convergence of a series $a_n$ using the Alternating series test, we need to satisfy the condition that $$|a_{n+1}| \leq |a_n|$$ and that $$\lim\limits_{n\rightarrow\infty} a_n = 0$$. ...
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### Absolutely improper integrability implies improper integrability

This was written in Freitag's (p196) A continuous function $f:[a,b) \rightarrow \mathbb{C}$ is called improperly integrable iff the limit $$\lim_{t\rightarrow b} \int_a^t f(x) \, dx$$ exists....
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### $\sum{\frac{(-1)^{n-1}}{{(n(n+1))^{1/3}}}}$ series convergence

$$\sum{\frac{(-1)^{n-1}}{{(n(n+1))^{1/3}}}}$$ Does this converge or diverge (absolute and/or conditional)?$\\$ I've tried Leibniz, D'Alembert, Cauchy and Cauchy-integral criteria, all that's left is ...
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### Absolute and conditional convergence of integral

I have the following improper integral: $$\int \limits_{1}^{\infty}\left(\cos\left(\frac{\sin(x)}{\sqrt{x}}\right) - \cos\left(\frac{\cos(x)}{\sqrt{x}}\right)\right)dx$$ And I need to figure out, ...
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### Absolute and Uniform Convergence of a Series

$$\sum_{k=1}^\infty \frac{(-1)^k}{\sqrt{k-1}+(-1)^k}$$ I put it on wolfram alpha and it shows that converges, but I think the analysis has to be different, because I need to know if the series ...
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### Absolute convergence of series from $-\infty \to \infty$

How would you show that the following series is absolutely convergent $$T\sum _{n=-\infty} ^\infty g(2\pi iTn)$$ where $$g(z) = {1 \over z^2-a^2}, \space \space \space a>0,$$ and $2 \pi i Tn$ ...
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### Absolute and Conditional Convergence of $\int_0^{+\infty}\left(x + \frac{1}{x}\right)^{\alpha}\sin (x^3) \mathrm{d}x$

Investigate the absolute and conditional convergence of the integral $$\int_0^{+\infty}f(x)\mathrm{d}x = \int_0^{+\infty}\left(x + \frac{1}{x}\right)^{\alpha}\sin (x^3) \mathrm{d}x$$ for all ...
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### Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$.

Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$. I know that we should use absolute convergence but not sure how that applies in this case.
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### How to figure out either series are absolute or conditional convergence $\sum_1^\infty \cos(\frac{\pi n}{3})(n^{\frac{1}{\sqrt[6]{n+6}}}-1)$

$$\sum_1^\infty \cos\left( \frac{\pi n} 3 \right) \left(n^{\frac 1 {\sqrt[6]{n+6}}}-1\right)$$ I tried to use Dirichlet, but unsuccessfully. Please, give me hints. Thanks a lot.
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### Where does this series converge $\sum_{n=1}^\infty\frac{(-1)^{\mu(n)}}{n^s}$, being $\mu(n)$ the Möbius function?

Let $\mu(n)$ the Möbius function and $s=\sigma+it$ the complex variable, then I've defined the Dirichlet series $$\epsilon(s):=\sum_{n=1}^\infty\frac{(-1)^{\mu(n)}}{n^s}.$$ And now I know that using ...
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### Does the comparison test imply absolute convergence?

As my textbook states it: Let $\Sigma a_n$ be a series where $a_n \ge 0$ for all $n$. (i) If $\Sigma a_n$ converges and $|b_n| \le a_n$ for all $n$, then $\Sigma b_n$ converges. (ii) ...