# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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### Convergence of integral: $\int_{0}^{\infty} \frac{x^{p}\sin{x}}{1+x^{q}}\ dx$

I have problem with determining about convergence of integral below.I firstly tried to determine about absolute convergence by limit test but i cannot find relation between parameter $p$ and $q$.For ...
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### Convergence of integral: $\int_{0}^{\infty} \frac{\sin{\left(x+\frac{1}{x}\right)}}{x^{a}}\ dx$

I need to determine about convergence of integral below.I found out that it doesn't converge absolutely but i have problem with determining about conditional convergence.I am not sure how to use ...
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### Convergence of: $\sum _{n=1}^{\infty }{\left(-1\right)^{n}\frac{2^{n}\sin^{2n}{\left(x\right)}}{n}}$

I solved that it converges absolutely on $\bigcup \left(-\frac{\pi}{4}+k\pi,\frac{\pi}{4}+k\pi\right) \text{for all k belongs to integer numbers}$ I don't know how to deal with "non-absolute" ...
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### Show that $2\cos z = e^{iz} + e^{-iz}$. Do I need absolute convergence?

I'm reading Conway's "Functions of one complex variable". Define $$e^z =\sum_{n=0}^\infty \frac{z^n}{n!}$$ $$\cos z =\sum_{n=0}^\infty \frac{(-1)^nz^{2n}}{(2n)!}$$ It is already shown that these ...
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### Is my understanding of the proof of rearrangement theorem correct?

I'm reading the proof of rearrangement theorem. Could you please verify if my understanding of the last part of the proof is correct or not? The authors said The inequality $(8.3)$ implies the ...
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### why converges absolutely needed for that prove *?*

Theorem: If $\sum_{k=1}^{\infty} a_k$ converges absolutely, then any rearrangement of this series converges to the same limit. I will try to prove without using converges absolutely condition. ...
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### Radius of convergence of power series and absolute convergence

We know that if the power series $\sum_{k=0}^\infty a_kz^k, a_k, z \in \mathbb{C}$ has a radius of convergence $R$ then it converges absolutely for $|z| < R$ and diverges for $|z| > R$ but the ...
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### Convergence and Divergence in $2^{-\ln n}$

The series that I need to check is $2^{-\ln n}$ and $3^{-\ln n}$. For the second case, after doing the preliminary test (which gave me zero so that I may proceed further), I tried to solve it by ...
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### Show that the next double power series is convergent

Show that $$\sum_{k=0}^{\infty}\sum_{n=0}^{\infty}[k|\beta +1]_n\frac{|t|^k}{k!}\frac{|x|^n}{n!}$$ is convergent $\forall t \in \mathbb{R}$, and $x, \beta \in \mathbb{R}$ such as $|\beta x|<1$, ...
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### Convergence of $\sum_{k=1}^{\infty}\frac{\cos(\theta k)}{\sqrt{k}}$

Say if the following series $$\sum_{k=1}^{\infty} \frac{\cos(\theta k)}{\sqrt{k}}$$ for $θ \in \mathbb{R}$ is convergent. Is it absolutely convergent? I don't know how to approach this problem. ...
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### Sum of the Riemann-Zeta function at the integers

If you consider the series of reciprocal positive integers (the harmonic series), the sum diverges. However, some subsets of this series, such as the reciprocal cubes, will sum to something finite. I ...
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### Absolute convergence of $\sum_{k=0}^{+\infty}a_{n}(x+2)^n$

Define the series $$\sum_{n=0}^{+\infty}a_{n}(x+2)^n.$$ All we know is that it converges in x=4. Why can't I say it converges absolutely in x=4?
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### If $\sum a_n$ converges absolutely, then so does $\sum f(a_n).$

Hi I am stuck on this problem. (please dont give the solution I just need some help to formalism my solution) Let f: R to R be differentiable, with continuous derivatives and f(0) = 0 If $\sum{a_{n}}$...
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### series convergence imply nx approaches 0

Proof: For a decreasing sequence of positive reals, show that if the sum converges, then $nx_n \to 0$ but the converse is not true The first part I just assumed a positive limit the series converge ...
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### Why does this inequality hold? (“Complex Analysis” by Kunihiko Kodaira.)

I am reading "Complex Analysis" by Kunihiko Kodaira. I cannot understand why the following inequality holds. Please tell me the reason why this inequality holds.
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### If every rearrangement of the series converges uniformly then the series converges absolutely uniformly

Let $I \subset \mathbb{R}$ and for $\forall n \in \mathbb{N}: f_n \in C(I, \mathbb{R})$. Prove that if for any $\sigma:\mathbb{N} \rightarrow \mathbb{N}$ bijection, the series $$\sum_n f_{\sigma(n)}$$ ...
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### The proof of Taylor's theorem

Studying Analysis with 'Principles of Mathematical Analysis Third edition' written by Walter Rudin, I got some trouble from the proof of Taylor's theorem. The theorem and the proof from the book are ...
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### Convergence when ratio test=1

When using the ratio test for absolute convergence of a series $\sum_{n=1}^\infty a_{n}$, if the limit of the ratio $$|a_{n+1}|/|a_{n}|=1$$ when $n \rightarrow \infty$, the fate of the series is ...
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### Find the convergence radius for this power series

The Problem: Find the convergence radius of $\sum_{n=0}^{\infty} \frac{n}{5^{n-1}} z^{\frac{(n)(n+1)}{2}}$ My attempts to find a solution I apply either the ratio test and end up with this ...
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### Help with convergence tests for series

I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so ...
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### Prove that $\sum_{i \in S} c_i \leq \sum_{i \in S'} c_i$ if $S \subset S' \subset \mathbb{N}$.

Is the following proof correct or not? Please tell me better proof. Let $\{c_n\}$ be a sequence of positive numbers such that $\sum c_n$ converges. Let $S \subset S' \subset \mathbb{N}$. ...
### About $f(x) = \sum_{x_n < x} c_n$ in Remark 4.31 on p.97 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.
I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. In Remark 4.31 on p.97, Rudin wrote this symbol $$\sum_{x_n < x} c_n.$$ What is the definition of this symbol ...