Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

1
vote
1answer
190 views

Does an unconditionally convergent series of complex numbers converge absolutely?

Suppose $\sum a_n$ is a series of complex numbers, if $\sum a_n$ and its every rearrangement all converge to the same sum, does $\sum a_n$ converge absolutely?
8
votes
1answer
206 views

Prove that $\sum_{n=0}^{\infty}{{a_n}{z^n}}$ converges absolutely and uniformly in $D$.

PROBLEM Suppose that the complex series $\displaystyle \sum_{n=0}^{\infty}{a_n}$ converges. Let $r < 1$ and set $D = \{z \in \mathbb{C} : |z| < r\}$. Prove that $\displaystyle \...
0
votes
3answers
359 views

Determine whether $\sum\limits_{n=1}^{\infty} (-1)^{n-1}(\frac{n}{n^2+1})$ is absolutely convergent, conditionally convergent, or divergent.

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$\sum_{n=1}^{\infty} (-1)^{n-1}\left(\frac{n}{n^2+1}\right)$$ Here's my work: $b_n = (\dfrac{n}{n^...
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 !
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 ...
0
votes
2answers
174 views

Prob. 9, Chap. 6, in Baby Rudin: Which one of these two improper integrals converges absolutely and which one does not?

Here are the links to three earlier posts of mine on Prob. 9, Chap. 6, in Baby Rudin here on Math SE. Prob. 9, Chap. 6, in Baby Rudin: Integration by parts for improper integrals Prob. 9, Chap. 6, ...
0
votes
0answers
57 views

Series of Cauchy Products

I am trying to figure out convergence conditions for the series $B\equiv \sum_{n=0}^\infty A_n A_n'$ where $A_n = \sum_{k=0}^\infty a_{n+k}$ and $A_n'$ being the transpose of $A_n$. I have a number ...
0
votes
2answers
96 views

In the alternating series test, what is the condition for convergence? [closed]

When testing for convergence of a series $a_n$ using the Alternating series test, we need to satisfy the condition that $$|a_{n+1}| \leq |a_n|$$ and that $$\lim\limits_{n\rightarrow\infty} a_n = 0$$. ...
0
votes
1answer
120 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....
2
votes
3answers
198 views

$\sum{\frac{(-1)^{n-1}}{{(n(n+1))^{1/3}}}}$ series convergence

$$\sum{\frac{(-1)^{n-1}}{{(n(n+1))^{1/3}}}}$$ Does this converge or diverge (absolute and/or conditional)?$\\$ I've tried Leibniz, D'Alembert, Cauchy and Cauchy-integral criteria, all that's left is ...
1
vote
0answers
105 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
vote
2answers
70 views

Convergence, absolute convergence, divergence of series

Let the series $\sum_{n=0}^\infty{\frac{a_n}{3^n}}$ be convergent, but the series $\sum_{n=0}^\infty{\frac{{(-1)^n}{a_n}}{3^n}} $ be divergent. Show whether: a)$\sum_{n=0}^\infty{\frac{a_n}{3^n}}$ ...
1
vote
2answers
55 views

Convergence of series $\sum_{n=1}^\infty \frac {cos(n \pi)}{(n+1)ln(n+1)} $

How can I figure out if this series converges absolutely? $$ \sum_{n=1}^\infty \frac {\cos(n \pi)}{(n+1)\ln(n+1)} $$ The ratio and root test are both inconclusive (according to Wolfram Alpha).
0
votes
0answers
227 views

On Mertens' theorem

I have a problem understanding the proof of Merten's theorem for the case $$\sum_{n=1}^{\infty}a_{n},\sum_{n=1}^{\infty}a_{n} - abs.convergent$$ then $ \sum_{n=1}^{\infty}c_{n}$ is convergent and $$\...
1
vote
1answer
133 views

Absolutely convergent series as sum of subseries

I was studying the notion of series, it states that a series $ \sum \limits_{s \in S} x_s $, in $\mathbb{R}$ over a set $S$, is convergent if there exists a $F\in \mathbb{R}$ such that for every $\...
0
votes
1answer
29 views

Rearrangements and unconditional convergence

Let us consider the convergente series: $∑_{n=1}^{∞}a_{n}=S$ where $S$ is a real number. I know about Rearrangements and unconditional convergence (https://en.wikipedia.org/wiki/Absolute_convergence). ...
2
votes
2answers
105 views

Convergence/Absolute Convergence for $\int_{0}^\infty{\sin^{n}\left(x\right)\over x}\,\mathrm{d}x $

Prove convergence and absolute convergence of $$ \int_{0}^{\infty}{\sin^{n}\left(x\right)\over x}\,\mathrm{d}x $$ I have seen a related question already, however the answers are not that helpful ...
0
votes
1answer
116 views

Check convergence of an improper Integral

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 $\...
0
votes
0answers
350 views

Absolute and conditional convergence of integral

I have the following improper integral: $$ \int \limits_{1}^{\infty}\left(\cos\left(\frac{\sin(x)}{\sqrt{x}}\right) - \cos\left(\frac{\cos(x)}{\sqrt{x}}\right)\right)dx $$ And I need to figure out, ...
1
vote
2answers
80 views

Absolute and Uniform Convergence of a Series

$$\sum_{k=1}^\infty \frac{(-1)^k}{\sqrt{k-1}+(-1)^k}$$ I put it on wolfram alpha and it shows that converges, but I think the analysis has to be different, because I need to know if the series ...
1
vote
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 $2 \pi i Tn$ ...
2
votes
0answers
81 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 ...
1
vote
1answer
145 views

Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$.

Find a Dirichlet series for $\frac{\zeta(s-1)}{\zeta(s)}$ valid for $Re(s)>2$. I know that we should use absolute convergence but not sure how that applies in this case.
1
vote
2answers
84 views

How to figure out either series are absolute or conditional convergence $\sum_1^\infty \cos(\frac{\pi n}{3})(n^{\frac{1}{\sqrt[6]{n+6}}}-1)$

$$\sum_1^\infty \cos\left( \frac{\pi n} 3 \right) \left(n^{\frac 1 {\sqrt[6]{n+6}}}-1\right)$$ I tried to use Dirichlet, but unsuccessfully. Please, give me hints. Thanks a lot.
1
vote
1answer
74 views

Why doesn't this series converge absolutely? Is it uniformly convergent?

$\sum_{k=1}^\infty \frac{(-1)^{k+1}}{x^2+\sqrt{k}}$ Why doesn't it converge absolutely? I know it converges pointwise by alternating series test. For uniform convergence: I tried approximating $|...
-1
votes
1answer
73 views

Convergence of series $\sum_{n =1}^{\infty}\sin(\frac{\pi\cdot n}{4})\cdot \sqrt[9]{\ln(\frac{n+12}{n+9})}$

$$\sum_{n =1}^{\infty}\sin\left(\frac{\pi\cdot n}{4}\right)\cdot \sqrt[9]{\ln\left(\frac{n+12}{n+9}\right)}$$ How to find convergence of this series? I researched the absolute convergence and get $$\...
0
votes
4answers
227 views

2.8.2 Stephen Abbott : Absolute convergence for double series

So here is the problem. Given absolute convergence for a double series (infinite sum over $|a_{ij}|$) , show the double series $(a_{ij})$ converges . The proof strategy is: 1) keep one index fixed - ...
2
votes
0answers
89 views

How to decide if it the series absolute convergent or conditional? [closed]

$$\sum_{n = 1}^{\infty}(-1)^{\frac{n(n+1)}{2}}(n^{\frac{1}{\sqrt[10]{n+10}}}-1)$$ Actually, I don't have idea how to do it. Help me please Thanks a lot.
1
vote
1answer
78 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
votes
3answers
41 views

what does this series do?(Converges or Diverges) [closed]

what this series do when n is going to be a large number. ln((2n+1)/(2n-1)) when n is tending to infinity? I used integral test, ratio test, comparison test and ... what do you think about this ...
0
votes
1answer
68 views

Radius of Convergence for (2-z)/(1-z) via the Ratio Test

I have a power series $\sum a_n z^n=\frac{2-z}{1-z}$ and I want to determine the radius of convergence of this power series. Rewriting the expression on the RHS we see that it is equivalent to $1+\...
0
votes
2answers
29 views

convergence of $\int\limits_{0}^{1}\frac{\sqrt{x}}{\sqrt{1-x^4}}\, dx $

I would like to prove that this integral is convergent. Although it's easy to evaluate it, I would like more ellegant way to do it. Thanks $$\int\limits_{0}^{1}\frac{\sqrt{x}}{\sqrt{1-x^4}}\, dx $$
-1
votes
1answer
38 views

Convergence of $\sum\limits_{n=1}^{\infty} k^{1/n}$

Does $\sum\limits_{n=1}^{\infty} k^{1/n}$ converge when $k<1$ ??? How to show whether it does or does not then? Integral test or comparison test with $k^n$ does not seem to work.
2
votes
1answer
81 views

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{...
4
votes
3answers
155 views

Does the series $\sum^\infty_{n=1} \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ diverge, converge, or converge absolutely?

Problem statement: Does the series $$\sum^\infty_{n=1} \frac{(-1)^n}{n^{1+\frac{1}{n}}}$$ diverge, converge, or converge absolutely? EDIT: I appreciate all the answers so far. In my book, we haven't ...
0
votes
1answer
51 views

Non-convergent rearrangement of the terms of a relatively convergent series

Let $\sum_{n=0}^{\infty}{x_{n}}$ be a series in $\mathbb{C}$ that is conditionally convergent (i.e. convergent, but not absolutely). Show that there exists a bijection $\pi : \mathbb{N} \rightarrow \...
0
votes
2answers
61 views

Convergence conditions for $\sum\limits_{n\ge1}(-1)^{n-1}\frac{(\log n)^p}n$ where $p\in\mathbb R$

As far as I know, the fact that the series starts at $n=1$ is intentional, so I know I can immediately omit the case where $p\le0$. I decided to use the alternating series test, so to show that the ...
0
votes
1answer
860 views

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 ...
1
vote
3answers
63 views

How to show $\sum a_n$ cannot converge absolutely if lim $(\sqrt{n} |a_n|) = L, L >0$

I'm trying to show that if lim $(\sqrt{n} |a_n|) = L, L >0$, then $\sum a_n$ cannot converge absolutely. I'm trying to work it from the definition of a limit that for any $\epsilon > 0$, there ...
0
votes
2answers
33 views

Convergence of a Particular Sequence

I was supposed to use the convergence tests to determine whether the following series converges, converges absolutely, or diverges. Almost every test I used was inconclusive, the series is : $$\sum_{...
1
vote
0answers
54 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 ...
0
votes
2answers
232 views

Proof regarding absolutely convergent series

Assume that $\sum_{n=1}^{\infty} c_n$ is absolutely convergent. Let $\phi$ : $\mathbb{N}$ → $\mathbb{N}$ be a bijection. Set $d_n = c_{\phi(n)}$ for $n\in \mathbb{N}.$ Show that $\sum_{n=1}^{\infty} ...
3
votes
2answers
95 views

Where does this series converge $\sum_{n=1}^\infty\frac{(-1)^{\mu(n)}}{n^s}$, being $\mu(n)$ the Möbius function?

Let $\mu(n)$ the Möbius function and $s=\sigma+it$ the complex variable, then I've defined the Dirichlet series $$\epsilon(s):=\sum_{n=1}^\infty\frac{(-1)^{\mu(n)}}{n^s}.$$ And now I know that using ...
0
votes
1answer
381 views

Does the comparison test imply absolute convergence?

As my textbook states it: Let $\Sigma a_n$ be a series where $a_n \ge 0$ for all $n $. (i) If $\Sigma a_n $ converges and $|b_n| \le a_n$ for all $n $, then $\Sigma b_n$ converges. (ii) ...
0
votes
2answers
46 views

Absolute Convergence and Supremum?

How would one prove the following? If $\sum_{n}a_{n}$ of non-negative terms is absolutely convergent and $\sum_{n}a_{n}=a$, then $a=\sup\{S\}$, where $S$ is the set of all finite sums of values of $...
1
vote
2answers
151 views

Intuitive understanding of the difference between absolute and conditional convergence for improper integrals

I am looking for an intuitive explanation that explains the difference between absolutely and conditionally convergent (Riemann) improper integrals. Please note I understand how one goes about ...
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
8answers
3k views

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 ...
3
votes
1answer
116 views

The definition of the complex function $\sum_{n=1}^{\infty}\frac{\mu(n)}{n}z^n$ on the open disk, where $\mu(n)$ is the Möbius function

I am interested if it is known, that is if was in the literature the following function $$\sum_{n=1}^{\infty}\frac{\mu(n)}{n}z^n$$ where $\mu(n)$ is the Möbius function and $z$ is the complex variable ...
1
vote
1answer
128 views

Ratio test for Series Convergence

I'm currently looking through the Wikipedia Article about the ratio test for convergence of a series. The article includes a decision diagram for the ratio test. The diagram look something like this: ...