# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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### $b_n$ bounded, $\sum a_n$ converges absolutely, then $\sum a_nb_n$ also

a) Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely. I wanted to use the comparison test to show it's true, but I think I ...
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### If every absolutely convergent series is convergent then $X$ is Banach

Show that A Normed Linear Space $X$ is a Banach Space iff every absolutely convergent series is convergent. My try: Let $X$ is a Banach Space .Let $\sum x_n$ be an absolutely convergent series ....
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### A series converges absolutely if and only if every subseries converges

Question: A subseries of the series $\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\sum _{n=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove that $\sum _{n=1}^\infty a_n$ ...
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### Arranging the alternating harmonic series to sum to $\sqrt{2}$

Since the alternating harmonic series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \frac11-\frac12+\frac13-\frac14+\cdots$$ is convergent but not absolutely convergent, any real number can be obtained ...
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### Does absolute convergence of a sum imply uniform convergence?

Suppose I have a series $\sum_{n = 0}^{\infty} f_{n}(x)$ which converges absolutely to a function $f(x)$. Does the series converge uniformly to $f(x)$? I want to say this follows from Dini's Theorem, ...
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### Does $\sum_{n=1}^\infty(-1)^n \sin \left( \frac{1}{n} \right)$ absolutely converge?

Does $$\sum_{n=1}^\infty(-1)^n \sin \left( \frac{1}{n} \right)$$ converge conditionally or absolutely? I know that this series converges conditionally using the Leibniz's convergence test, but what ...
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### Identity between absolutely convergent generalized Dirichlet series

The following identity between generalized Dirichlet series, both absolutely convergent in the whole complex plane, occurs when investigating functions of the Selberg class of degree $d=0$: \begin{...
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### Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
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### Why is $\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolut convergent?

Why is $\sin(tA)=\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolute convergent ? for a $n\times n$ real matrix $A$ and $t\in \mathbf R$ Which crieterion is to use ?
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### Grouping the Summation

Let $a_n \in \mathbb{C}$ and consider $\sum a_n$ and grouping as $\sum (a_n + a_{n+1})$. Under what assumptions we can claim absolute convergence of grouped sum implies convergence of the original sum?...
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### Definition for the complex exponential function

We define the exponential function as $\exp(z)=\sum\limits_{j=0}^\infty= \frac{z^j}{j!}$ for all $z\in \mathbb{C}$. Lets now compute $\exp(0)$, then we would have to calculate $0^0$ which undefined. ...
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### On absolutely convergent series $\sum _{n=1}^{\infty}a_n$ such that $\sum _{n=1}^{\infty}a_{kn}=0,\forall k \ge 1$

Let $\sum _{n=1}^{\infty}a_n$ be an absolutely convergent series of real terms such that $\sum _{n=1}^{\infty}a_{kn}=0,\forall k \ge 1$ . For $m,n\in\mathbb N , S_n(m):=\sum a_{mk}$ , where the sum ...
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### Testing convergence of series $\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$

Lets have a series $$\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$$ However, I have absolutely no clue how to try to continue. I could probably use the integral criterion and ...
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Let $A$ be a countably infinite set, and let $f:A\to\mathbb{R}$ be a function that takes elements of $A$ to the reals. Suppose that $\sum_{w\in A} f(w)$ is well-defined (see note below). Also suppose $... 4answers 148 views ### Series and Sequences - Absolute Convergence I'm practicing for my final exams this week but the past year papers have no answers so I'm not sure if my answers are acceptable, was hoping someone would look at my proof and let me know if it is a ... 1answer 78 views ### Prob. 12, Chap. 3 in Baby Rudin: Some results involving the remainder of a convergent series of positive term series Here is Prob. 12, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose$a_n > 0$and$\sum a_n$converges. Put $$r_n = \sum_{m=n}^\infty a_m.$$ ... 1answer 63 views ### Does the series corresponding to a Cauchy sequence **always** converge absolutely? Let$X$be a normed vector space and consider a Cauchy sequence$(x_n)_{n\in\mathbb{N}}$in$X$. Is it true that the corresponding series of our Cauchy sequence,$\sum_{i=1}^\infty x_i$, always ... 1answer 125 views ### 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.... 0answers 46 views ### Does convergence of$\Gamma(x)$imply convergence of$\Gamma(z)$? Is it generalisable? If we have the gamma function in integral form$\Gamma(x)=\int_\limits0^\infty e^{-t}t^{x-1}dt$and have proven that it converges for real$x>0$(to a point), can we then immediately conclude that$...
Check for convergence and absolut convergence. $$\int_{0}^{\infty}{\sin^{n}\left(x\right)\over x}\,\mathrm{d}x$$ I know how to do it if $n = 1$ but i dont understand the integration by parts of \$\...