Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

16
votes
1answer
5k views

$b_n$ bounded, $\sum a_n$ converges absolutely, then $\sum a_nb_n$ also

a) Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely. I wanted to use the comparison test to show it's true, but I think I ...
12
votes
1answer
344 views

Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in ...
11
votes
2answers
207 views

Absolute convergence when all the rotated series converge

The question here might be standard in some textbook. Let $a_n, n\ge1$ be a series of real numbers. It is evident that if $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$, then $\displaystyle \sum_{n\...
10
votes
3answers
8k views

Does absolute convergence of a sum imply uniform convergence?

Suppose I have a series $\sum_{n = 0}^{\infty} f_{n}(x)$ which converges absolutely to a function $f(x)$. Does the series converge uniformly to $f(x)$? I want to say this follows from Dini's Theorem, ...
10
votes
2answers
508 views

Series convergent but not absolutely? $\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$

For which real numbers $p>0$ does the series $$\sum_{n=1}^{\infty} \frac{\cos(n^p \pi)}{n^p}$$ converge? Obviously it converges absolutely for $p>1$ but what about $0<p<1$? I have the ...
10
votes
3answers
4k views

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 ....
8
votes
1answer
4k views

A series converges absolutely if and only if every subseries converges

Question: A subseries of the series $\sum _{n=1}^\infty a_n$ is defined to be a series of the form $\sum _{n=1}^\infty a_{n_k}$, for $n_k \subseteq \Bbb N$. Prove that $\sum _{n=1}^\infty a_n$ ...
8
votes
1answer
208 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 \...
7
votes
4answers
171 views

Find an example of series which converges only absolutely on $\mathbb Q$

I am currently working on the completeness of metric spaces, so I studied the following theorem: If $E$ is a Banach space then any absolutely convergent series is convergent. Since $\mathbb Q$ is ...
7
votes
2answers
322 views

Using the root test when the limit does not exist

I used the root test for the series $$ \sum_{n=1}^{\infty} \left(\frac{\cos n}{2}\right)^n. $$ I showed that $$ 0 \le \left|\frac{\cos(n)}{2}\right| \le \frac{1}{2} \implies \lim_{n\to\infty}\left|\...
7
votes
0answers
223 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 ...
7
votes
0answers
102 views

$\sum_{n=1}^{\infty} a_n$ converges absolutely and $\sum _{n=1}^\infty a_{kn}=0 ,\forall k \ge 1 $ ; then $a_n=0 , \forall n \in \mathbb N $? [duplicate]

Suppose that the series $\sum_{n=1}^{\infty} a_n$ of real terms converges absolutely and $\sum _{n=1}^\infty a_{kn}=0 ,\forall k \in \mathbb N $ , then how to prove that $a_n=0 , \forall n \in \...
6
votes
5answers
661 views

Arranging the alternating harmonic series to sum to $\sqrt{2}$

Since the alternating harmonic series $$ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \frac11-\frac12+\frac13-\frac14+\cdots $$ is convergent but not absolutely convergent, any real number can be obtained ...
6
votes
0answers
311 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 ...
5
votes
3answers
227 views

Is it possible that $\sum n a_n^2$ converges but $\sum a_n$ diverges?

Yesterday I was thinking about a problem, when an interesting question appeared: Does there exist a sequence $a_n \geq 0$ of non-negative real numbers such that $\sum_{n \geq 1} n a_n^2 < \infty$ ...
5
votes
3answers
78 views

Find the interval of convergence of $x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 +…$

How to find the interval of convergence of the following series: $x + \frac{1}{2} x^2 + 3x^3 + \frac{1}{4}x^4 +...$ I have no idea what to proceed. Any help? Thanks!
5
votes
1answer
603 views

Show that if $(\sum x_n)$ converges absolutely and $(y_n)$ is bounded then $(\sum x_n y_n)$ converges

This is the exercise 2.7.6 of the book Understanding analysis of Abbott, I want a check of my proof and if is needed additional information to complete it. a) Show that if the sequence $(\sum x_n)$ ...
5
votes
1answer
62 views

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 $$\sum_n f_{\sigma(n)}$$ ...
5
votes
1answer
106 views

$\sum_{i=1}^{\infty}a_n*b_n $ converges for all $\lim_{n \rightarrow \infty}b_n = 1$, show that $a$ converges absolutely

So this is given: $\sum_{n=1}^{\infty}a_n*b_n $ converges for all sequences $(b_n)$, such that $\lim_{n \rightarrow \infty}b_n = 1$. Somehow it should be showable that $(a_n)\,$converges absolutely, ...
4
votes
3answers
813 views

Why is the convergence absolute?

There is one thing my book uses in a proof after Abels theorem which I do not understand: Lets say that $\sum_{n=0}^\infty a_n$ converges. For $0\le x<1$, we look at $\sum_{n=0}^\infty a_n x^n$. ...
4
votes
1answer
338 views

If $\sum a_n$ is convergent, is $\sum\frac{a_n}{n}$ absolutely convergent?

Assume that a series $\sum\limits_{n\geqslant1} a_n$ is convergent. Does this imply that $\sum\limits_{n\geqslant1}\frac{a_n}{n}$ is absolutely convergent? My first thought is no, but I'm having a ...
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 ...
4
votes
6answers
134 views

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

Does the following sum converge? Does it converge absolutely? $$\sum_{n=1}^\infty \left(\sin \frac{1}{2n} - \sin \frac{1}{2n+1}\right)$$ I promise this is the last one for today: Using Simpson'...
4
votes
1answer
2k views

An absolutely convergent series may be rearranged.

Any rearrangement of an absolutely convergent sequence $(a_n)$ is another absolutely convergent sequence with the same limit. Let $(a_{\sigma(n)})$ be the rearranged sequence under the bijection of ...
4
votes
2answers
58 views

Absolute convergence of $\sum_{n=1}^{\infty}z_n^2$

There are two complex series given ($z \in \mathbb{C}$. We know that: $$\sum_{n=1}^{\infty}z_n^2$$ converges absolutely. We are to show that $$\sum_{n=1}^{\infty} \frac{z_n}{n}$$ also converges ...
4
votes
2answers
144 views

Question on a Proof of Rearrangements for Absolutely Converging Double Series

In Appendix B of Jameson's The Prime Number Theorem, the author gives a proof of the assertion that given real numbers $\left\{a_{j,k}\right\}_{j,k\ge 1},$ if $$\sum_{j=1}^\infty \sum_{k=1}^\infty \...
4
votes
2answers
103 views

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. ...
4
votes
1answer
150 views

Is there any popular name for this theorem in the standard literature?

Let $X$ be a normed space. Then $X$ is a Banach space if and only if the absolute convergence of any series in $X$ implies the conditional convergence of that series. Is there any name given to the ...
4
votes
1answer
170 views

Prob. 11 (d) in Baby Rudin: Given $a_n > 0$, is this condition also sufficient for divergence of $\sum \frac{a_n}{1+na_n}$?

Here's Prob. 11 (d), Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $a_n > 0$ and that the series $\sum a_n$ is divergent. Then what can be said ...
4
votes
1answer
81 views

Show that $\sum_{n = 1}^{\infty} \frac{1}{z + n} + \frac{1}{z-n}$ is absolutely convergent for all $z \in \mathbb{H}$

I need to show that the series $$ \sum_{n = 1}^{\infty} \frac{1}{z + n} + \frac{1}{z-n} $$ is absolutely convergent for all $z$ in the complex upper half plane $\mathbb{H} = \{ z \in \mathbb{C} : \Im(...
4
votes
1answer
109 views

Splitting an infinite unordered sum (both directions)

This question is a follow-up to this. Let $A, B_1, B_2, \dots$ be countable sets such that $\bigcup_{i \in \mathbb{N}}B_i = A$ and the $B_i$'s are disjoint. Let $f : A \to \mathbb{R}$ take elements ...
4
votes
1answer
109 views

Showing the following function is entire…

The full problem asks about the following function using it's Maclaurin series: $$f(x)=\left\{ \begin{array}{lr} \frac{\sin(z)}{z} & : z \neq 0\\ \;\;\;\;1 & : z=0 \end{array} \right.$$ I've ...
3
votes
6answers
174 views

Prove that $\sum\limits_{n=0}^{\infty}{(e^{b_n}-1)}$ converges, given that $\sum\limits_{n=0}^{\infty}{b_n}$ converges absolutely.

It's a question from a test that I had, and I don't know how to prove this, so I am forwarding this to you. $\sum \limits_{n=0}^{\infty }\:b_n$ is absolutely convergent series . How to prove that ...
3
votes
4answers
248 views

Series convergence or divergence?

I'm having a hard time determining if the following series converges (absolutely?) or diverges: $$\sum_{i=1}^n \frac 1 {2+\sin n}$$ I would really appreciate some help here. Thanks!
3
votes
5answers
236 views

Is the series $\sum _{n=1}^{\infty } (-1)^n / {n^2}$ convergent or absolutely convergent?

Is this series convergent or absolutely convergent? $$\sum _{n=1}^{\infty }\:(-1)^n \frac {1} {n^2}$$ Attempt: I got this using Ratio Test: $$\lim_{n \to \infty} \frac{n^2}{(n+1)^2}$$
3
votes
2answers
168 views

Prove that the series $\sum\limits_{n=-\infty}^{+\infty}f(x+n)$ converges absolutely for a.e. $x \in \mathbb{R}.$

Problem: Let $f$ be a Lebesgue integrable function on $\mathbb{R}.$ Prove that the series $$\sum\limits_{n=-\infty}^{+\infty}f(x+n)$$ converges absolutely for a.e. $x \in \mathbb{R}.$ What ...
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 ...
3
votes
1answer
425 views

Show the series $a_n/(1+a_n)$ converges absolutely

Given that the series $(a_n)$ converges absolutely. Show that the series $(\frac{a_n}{1 + a_n})$ converges absolutely. I am not really sure where to start. Any help would be great.
3
votes
1answer
52 views

Absolute convergence of $ \sum a_nx_n $ implies absolute convergence of $ \sum a_n$

I'm trying to find a proof (or a conunter example, but I'm somehow convinced that the statement is true) for the following fact: $$ \forall_{(x_n)_{n=1}^{\infty} \lim{x_n} = 0 } \sum\limits_{n=1}^{\...
3
votes
1answer
42 views

Is $\ell^1$ complete with this norm?

For $x \in \ell^1$ we set $\Vert x\Vert = \sup\limits_{N \in \mathbb{N}}|\sum\limits_{n=1}^{N}x_n|$. One can easily see that this is a norm on $\ell^1$. I was wondering if this space is now complete. ...
3
votes
2answers
54 views

Find absolute convergence of $\sum_{n=1}^{\infty}\left[{\frac{\sin{\frac 1 n}+\cos{\left({n\pi}\right)}}{n}}\right]$

I have the following series: $$\sum_{n=1}^{\infty}\left[{\frac{\sin{\frac 1 n}+\cos{\left({n\pi}\right)}}{n}}\right]$$ My textbook asks to determine the simple and absolute convergence of the series....
3
votes
1answer
66 views

Four (simple) questions on real number series

I'm trying some exercises on real number series, in which I have to see if the series are convergent or not: $$a) \sum_{n=1}^\infty \ (-1)^n \frac{2n+1}{3^n}$$ $$b) \sum_{n=0}^\infty \ (-1)^{n+1} \...
3
votes
2answers
1k views

$ \sum a_n$ converges absolutely if $\sum a_nb_n $ converges absolutely for all bounded $\{b_n\}$

My question is Let $\sum a_n $ be a series such that $\sum a_nb_n$ converges absolutely for every sequence $\{b_n\}$ that is bounded. Prove that $\sum a_n$ is absolutely convergent. My ...
3
votes
2answers
6k views

Absolute convergence of $\sin(n)/(n^2)$

Prove that $$\sum_{n=1}^{\infty} \frac{\sin(n)}{{n}^{2}}$$ is either absolutely convergent, conditionally convergent or divergent. Note that $$\sin(n) \in [-1,1] \text { for} \left| \sum_{n=1}^{\...
3
votes
1answer
2k views

Is the Series Sin(k)/k Absolutely Convergent?

I know that the series $\sum_{k=1}^\infty \frac{\sin k}{k}$ converges (to $\frac{\pi - 1}{2}$), though by crazy stuff with Dirichlet Kernels or by reverse-engineering $\frac{\pi - x}{2} = \sum \frac{\...
3
votes
2answers
256 views

Absolute and conditional convergence of function series

I have a problem. I have to explore absolute and conditional convergence of this function series Unfortunately. I didn't find in my reference any words about "absolute and conditional", Instead I've ...
3
votes
3answers
73 views

Does series$\sum_{n=0}^\infty (-1)^n {(2n)!\over 4^nn!(n+1)!}$ converges absolutely or conditionally?

I wonder the convergence of the series $$\sum_{n=1}^\infty (-1)^n {(2n)!\over 4^nn!(n+1)!}.$$ We can derive that the general term is equal to $$(-1)^n \times {1\over2} \times {3 \over 4} \times \...
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 ...
3
votes
1answer
61 views

Two problems on real number series

Consider the series: $$a) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\sin\frac{1}{2k})}$$ $$b) \sum_{n=1}^\infty \frac{a^n}{ \prod_{k=1}^n \ (1+\tan\frac{3}{2k})}$$ Showing that these two are ...
3
votes
2answers
629 views

Normal convergence implies uniform absolute convergence but not the other way round

How do I show that normal convergence of a series implies uniform and absolute convergence? So, a series $f_1+f_2+...$ of functions $f_n:D\rightarrow\mathbb{C}, D\subset\mathbb{C}$ is normally ...