# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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### If $\sum_{n=1}^{\infty}a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty}(\frac{n+1}{n})a_n$ is also absolutely convergent? [on hold]

If $\sum_{n=1}^{\infty}a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty}(\frac{n+1}{n})a_n$ is also absolutely convergent. I need converse examples if exists such that above assertion is false....
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### Divergence of power series

In my lectures we considered a power series and used the ratio test for absolute convergence to find the radius or interval of convergence. However it was also stated in my lectures that "we have ...
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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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### 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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### 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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### If $\sum\limits_{n=1}^∞u_n^2$ is convergent, then $\sum\limits_{n=1}^∞\frac{u_n}n$ is absolutely convergent [duplicate]

If $\{u_n\}$ is a sequence of real numbers and the series $\displaystyle\sum_{n=1}^{\infty}u_n^2$ is convergent, prove that the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is absolutely ...
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### $\sum u_v$ converges absolutely iff $\sum \log(1 + u_v)$ converges absolutely

I have trouble understanding the following proof of a fact in complex analysis. Assume $(u_v)_{v\geq1}$ is a sequence of complex numbers and $(u_v) \neq -1$ for all $v$. Then we have the following ...
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### Does this series converge absolutely $\sum_{n=1}^{\infty}\frac{b^{n}_{n}\cos(n\pi)}{n}$

Let $\{b_n\}$ be a sequence of positive numbers that converges to $\frac{1}{2}.$ Determine whether the given series is absolutely convergent. $$\sum_{n=1}^{\infty}\frac{b^{n}_{n}\cos(n\pi)}{n}$$ ...
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### Converge Test on the series $\sum \limits_{n=0}^{\infty} \left(\frac{2n+n^3}{3-4n}\right)^n$

I want to show, that $a:=\sum \limits_{n=0}^{\infty} \left(\dfrac{2n+n^3}{3-4n}\right)^n$ is not converging, because $\lim \limits_{n \to \infty}(a)\neq 0 \; (*)$. Therefore, the series can't be ...
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### Theorem 3.55 in Baby Rudin: How to make sense of the proof?

Here's Theorem 3.55 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. If $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement ...
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### Convergence of $a_k=\sum_{k=5}^{+\infty}(-1)^k({3\over2})^{-k}(k^2+5)\sin(k+5)$

Firstly, $\lim_{k \to +\infty}a_k=0$,so necessary condition for convergence is satisfied.If we start to study absolute convergence we have : a_k=({3\over2})^{-k}(k^2+5)|\sin(k+5)|\leq({3\over2})^{-k}...
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### If $a_n$ doesn't have any subsequence that converges, can $|a_n|$ converge?

If a sequence $a_n$, $n\in\mathbb{N}$ doesn't have any convergent subsequence can $|a_n|\rightarrow a$, $a\in[0,\infty)$? My intuition says that this isn't possible but I'm not sure how to prove it..
I'm stuck in the proof of the following theorem of complex analysis: There is a $R\in\mathbb{R}_{0}^{+}\cup\{\infty\}$ s.t.: (i) $\sum_{k=0}^{\infty} a_k (z-z_0)^k$ converges absolutely and ...
Background Let $\left\{ {{a_n}} \right\}_{ - \infty }^\infty$ be a two sided sequence (is there a more proper term?) of complex numbers. As far as I know (please correct me if I am wrong) we say ...