# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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### For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

Unless I'm making a mistake, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely then the ...
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### Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
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### Two identities involving Gregory coefficients and different arithmetic functions from an integral representation

This morning I've deduced two identities that involve Gregory coefficients $G_n$ invoking the so-called Schröder's integral formula (this is the Wikipedia's article for Gregory coefficients). The ...
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### Showing that $(a_n)_n \in l_1$ provided $\sum_{k=1}^\infty a_kx_k$ exists for any $(x_n)_n \in c_0$

I tried first using the fact that $c_0$ is Banach to apply the Uniform Boundedness Principle on the function series $(T_n)_n = \{\sum_{k=1}^n a_kx_k\}$, $T_n:c_0 \rightarrow \mathbb{K}$, and then to ...
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### Theorem 3.55 in Baby Rudin: Every re-arrangement of an absolutely convergent series converges to the same sum in every normed space?

Here's Theorem 3.55 in the book Principles of Mathematical Analysis by Walter Rudin, third edition. If $\sum a_n$ is a series of complex numbers which converges absolutely, then every ...
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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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### Determine Convergence of $\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$

I know that $\frac{n!}{(n+1)!}$ can be reduced to $\frac{1}{n+1}$, but i'm not sure about this one. $$\sum_{n=1}^\infty \frac{(-1)^nn!}{(n+100)!}$$ In my notes, my professor reduced it to a p-...
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### System stability and $z$-transform

I have been struggling to understand the relation between $z$-transform and the study of analysis but there is something that puzzles me. Stability of a discrete time system is happens if and only if ...
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### On a variation of a problem posted by The Lviv Scottish Book in MathOverflow, using the Möbius function

Few days ago I've known (but currently I am not able to understand the answer) a nice problem proposed in MathWoverflow by the user Lviv Scottish Book, that is [1]. Yesterday using Wolfram Alpha ...
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### Studying the convergence and absolute convergence of $\sum_{n=1}^{\infty} cos(2^n)$.

$$\sum_{n=1}^{\infty} cos(2^n)$$I have tried to use several convergence test without any results. Also, since $a_n=cos(2^n)$ always take positive values, $|a_n|=a_n$ and therefore if $a_n$ converges ...
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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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### 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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### Prove that the series $\sum\limits_{k=1}^{\infty}[\ln(ak+b)- \ln(ak)]$ diverges

Let a and b be positive numbers. Prove that the series $\sum_{k=1}^{\infty}(ln(ak+b)- ln(ak))$ diverges. At first I thought expanding it would mean a few terms get cancelled out but it only works ...
387 views

### Eisenstein series converge absolutely for $k\geq 2$

I am looking at Eistenstein series on modular forms: https://en.wikipedia.org/wiki/Eisenstein_series The page claims that the series converges absolutely to a holomorphic function of τ when $k\geq 2$....
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### Stochastic Convergence

I need help figuring out if a series of "apparently random" digits are the result of the same (possibly non-polynomial) function, ergo, not-random but deterministic. The highest level math I know is ...
133 views

### Test convergence of the integral $\int_0 ^{+ \infty} t^{-\alpha} e ^ {-\cos t} \cdot \sin (t ^ \beta)dt$

I'm given the integral $$\int_0 ^{+ \infty} \frac{e ^ {-\cos t} \cdot \sin (t ^ \beta)}{t^\alpha} dt \qquad a,b \in \mathbb{R}$$ and I need to test the absolute convergence. I split it in two parts, ...
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### Bounding support of a probability measure by calculating radius of convergence of Stieltjes transform given by a sum

The general idea It is a well-known fact that the Stieltjes transform $s_\mu$ of a probability measure $\mu$ on $\mathbb R$ is analytic on $\mathbb C\setminus\text{supp}(\mu)$. So if I wanted to ...
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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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### Evaluating $-\ln(1-x)$ and an infinite sum

I did a little evaluation of the function $-\ln(1-x)$ with a sum of an infinite series, but I have come to a contradiction, and I would like to ask your help to find my mistake. Let us start with the ...
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### Convergence of $\sum\limits_{n=1}^{\infty} {\dfrac{(-1)^{n-1}} {(\sqrt{n}+(-1)^{n-1})^p}}$

Find $p$ that makes $\sum\limits_{n=1}^{\infty} {\dfrac{(-1)^{n-1}}{(\sqrt{n}+(-1)^{n-1})^p}}$ converges. Which $p$ makes the series converges absolutely? I think that it converges for $p>0$, can ...
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### Kummer's transformation applied to a series involving the Möbius function: were rights my deductions and how get an idea of the improvement?

After I've read Kummer's recipe to get the acceleration of a series, I want to do an example related with the Möbius function $\mu(n)$. Question. I present my calculations, please A) I would ...
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### On the integral representation of the function $b(s)$ defining $b(n)=\operatorname{Bernoulli}_{2n}$, and Riemann's trick

I don't know if this approach was in the literature, I would like to know some expression for $\Re s>1$ of the product $$\zeta(s)b(s),$$ where $\zeta(s)$ is the Riemann Zeta function, and $b(s)$ is ...
127 views

### Prove absolute and uniform convergence on compact subsets of $\mathbb{C}$

Consider the series $$\sum_{n=-\infty}^{\infty} e^{\pi i [n^2 \tau + 2n z]},$$ where Im$(\tau) >0$ and $z \in \mathbb{C}$. Prove that this series converges absolutely and uniformly on compact ...
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### Convergent Series and Rearrangements

Given a convergent positive series, I have to prove that $$\sum^{\infty}_{n=1} a_n=\sum^{\infty}_{k=0}a_{2k+1} + \sum^{\infty}_{n=1} a_{2k}$$ which means that the sum of odd terms and the even terms ...
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### Infinite product convergence in complex analysis: $\prod (1+a_k)$ and $\prod|(1+ a_k)|$

If $\prod (1+a_k)$ converges, then to prove that $\prod|(1+ a_k)|$ converges. Any suggestion and idea about this will be highly appreciated. Thank you.
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### Prove that the sum of a series is differentiable

Prove that the series $$\sum_{k=2}^∞ \sin (kx)/k\ln^2(k)$$ is absolutely and uniformly convergent on $\mathbb{R}$. If the sum of the series is denoted by $f(x)$ prove that $f$ is differentiable at ...
### Divergence of $\sum_{n=0}^\infty |\sin \omega n|$
I'm looking for a simple argument to show that $$\sum_{n=0}^\infty |\sin \omega n|$$ does not converge for $\omega \neq k \pi$, $k \in \mathbb{Z}$. If $\omega = \frac{1}{b}\pi$ with \$b \in \...