Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

84 questions with no upvoted or accepted answers
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7
votes
0answers
225 views

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 ...
6
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0answers
315 views

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 ...
3
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0answers
58 views

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 ...
3
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0answers
71 views

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 ...
3
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0answers
265 views

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 ...
2
votes
1answer
23 views

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?...
2
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0answers
86 views

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-...
2
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0answers
26 views

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 ...
2
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0answers
100 views

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 ...
2
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0answers
54 views

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 ...
2
votes
0answers
78 views

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 ...
2
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0answers
82 views

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 ...
2
votes
1answer
60 views

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 ...
2
votes
2answers
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 $k\geq 2$....
2
votes
1answer
40 views

Convergence of $\sum_{n=0}^\infty \left (\frac{(-1)^n}{n+1}+\frac{(-1)^{n+1}}{n+2} \right )$

What would you say about this series about convergence and absolute convergence? $$\sum_{n=0}^\infty \left (\frac{(-1)^n}{n+1}+\frac{(-1)^{n+1}}{n+2} \right )$$ In use with: $$\sum_{n=1}^∞ \frac{1}{...
2
votes
2answers
219 views

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 ...
2
votes
0answers
381 views

Absolute convergence and continuity imply uniform convergence of Fourier series to the function?

I encounter the following problem: Let $f$ be a periodic continuous function in $[0,2\pi]$ such that the Fourier of $f$ is absolute convergence, that is $$|a_0|+\sum_{n=1}^\infty (|a_n|+|b_n|)<\...
2
votes
0answers
126 views

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 ...
2
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0answers
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, ...
2
votes
0answers
80 views

Absolute convergence implies convergence .

Absolute convergence implies that $ \mid \mid s_m\mid −\mid s_n \mid \mid \space = \space \mid \mid x_{m}\mid +\mid x_{m−1}\mid +…\mid x_{n+1} \mid \mid \space \leq \space ϵ $ But $\mid s_m −...
2
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0answers
187 views

A problem in the Hecke's trick method

In his 'Introduction to modular forms', Don Zagier deals with the Hecke's trick which I don't really understand : Let $$G_2(\tau)=-\frac{1}{24}+\sum_{n=1}^{+\infty}{\sigma_1(n)q^n}$$ and $$G_2^*(\...
2
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0answers
109 views

Sufficient condition for absolute convergence of series

I want to prove the following statement If $\sum_{n\in I} a_n$ converges with any rearrangements of a countable index set $I$, then $\sum_{n\in I} a_n$ is absolutely convergent. The finite case is ...
1
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0answers
45 views

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 ...
1
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0answers
27 views

On the convergence of a two-sided series

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 ...
1
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0answers
22 views

Limit of Coefficient Sequence and Radius of Convergence

Show that if the power series $\sum_{n=0}^\infty a_n x^n$ has radius of convergence $R$ and if $\lim_{n \to \infty} |a_{n+1}/a_n|$ exists, then the value of this limit is $R$ I think there might be a ...
1
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1answer
57 views

Examples of complete families of functions forming an absolutely convergent series

I am searching for some examples of complete families of functions $\left\{ \phi_m(t) \right\}_{m = 1}^\infty$ on $t \in [0, T]$ that form an absolutely convergent series: $$ \sum_{m = 1}^\infty |\...
1
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0answers
62 views

Convergence of sum on lattice in $\mathbb{C}$

I am trying to understand why the following sum converges $$\sum_{\lambda \in \Lambda\backslash\{0\}}\frac{1}{|\lambda|^3},$$ where $\Lambda=\{m+n\tau \mid m,n\in \mathbb{Z} \}$ with $\tau \in \...
1
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0answers
32 views

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 ...
1
vote
1answer
83 views

Prove absolute convergence of $\sum_{n=1}^{\infty} \frac{2^nn!}{(z+1)(z+3)…(z+2n+1)}$

I need to prove that $$\sum_{n=1}^{\infty} \frac{2^nn!}{(z+1)(z+3)...(z+2n+1)}, \operatorname{Re}z \gt \frac{1}{2}$$ is absolutely convergent. I've tried to use d'Alambert ratio test, but I got $$...
1
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0answers
31 views

Radius of convergence of this power series?

The coefficient of $(x^2)^n$ is $(n!)^2/(2n)!$. I did the problem and calculated the radius of convergence to be 2. But the book says it's 4. I'm having the same issue with similar problems where only ...
1
vote
1answer
40 views

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 !
1
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0answers
108 views

How to prove a certain integral is convergent, but not absolutely convergent?

Been struggling with this problem for quite some time now and can't seem to be able to find the solution by myself. The said integral is $$ \int\limits_{0}^{+\infty} \sin(x\ln^{1/3}(x))\ \mathrm{d}x ...
1
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0answers
32 views

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 $2 \pi i Tn$ ...
1
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1answer
79 views

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 ...
1
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0answers
55 views

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 ...
1
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0answers
136 views

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 ...
1
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0answers
24 views

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 ...
1
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0answers
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 ...
1
vote
1answer
50 views

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 ...
1
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0answers
112 views

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.
1
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0answers
132 views

Dirichlet series with abscissa of absolute convergence $= \frac{1}{2}$

I'm trying to figure out a Dirichlet series which has its abscissa of absolute convergence $=\frac{1}{2}$. I've been trying to think about using the formula for this abscissa: $$\sigma_a=\limsup_{n\...
1
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0answers
207 views

Rearrangement of an absolutely convergent series

Suppose I have the sequences $a_n$ such that $\sum_{n=1}^\infty a_n$ converges absolutely. Take the following rearrangement: $(a_1+a_3+....)+(a_2+a_4+....)$. The question is if that is a valid ...
1
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0answers
47 views

Type of convergence: absolute convergence vs convergence of partial sums

Kolmogorov's 3-series theorem states that if $X_k, k \geq 1$ are independent, $A > 0$ a threshold level, $Y_k = X_k 1(|X_k|\leq A)$, then $\sum_k X_k$ converges iff all three series converge: $\...
1
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0answers
188 views

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 ...
1
vote
0answers
96 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
1
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0answers
41 views

Is any of this true about infinite series of functions?

Let $f_n^+(x)$ be a sequence of non-negative functions $f_n^+: X \to \Bbb{R}_{\geq 0}$, such that each $f_n^+$ has countably many zeros. Then if $f(x) = \sum f_n^+(x)$ converges point-wise, the ...
0
votes
0answers
16 views

Absolute convergence of series variant of the geometric series

So I want to prove whether the following series converges absolutely or not: $$\frac{1}{2}\sum_{n=0}^\infty (n^2+3n+2)q^n$$ where $ q \in \mathbb{C}, \mid q\mid<1.$ My attempt was: $$\frac{1}{2}...
0
votes
0answers
33 views

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 ...
0
votes
0answers
22 views

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 \...
0
votes
0answers
39 views

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 ...