Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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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 ...
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 $\sum _{n=1}^\infty a_n$ ...
6
votes
5answers
716 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 ...
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, ...
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{\...
-2
votes
1answer
101 views

Show that $\sum_{n = 1}^\infty n^qx^n$ is absolutely convergent, and that $\lim_{n \rightarrow \infty}$ $n^qx^n = 0$

I'm having trouble with proving the following for my math study: Let $x$ be a real number with $|x| < 1$, and $q$ be a real number. Show that the series $\sum_{n = 1}^\infty n^qx^n$ is absolutely ...
4
votes
1answer
111 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 ...
3
votes
1answer
441 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
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
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 ...
1
vote
2answers
154 views

Testing convergence of series $\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$

Lets have this problem. $$\sum_{n=3}^\infty\frac{1}{n (\ln(n))^p(\ln\ln(n))^q}$$ I have rewritten this to a form $$\sum\frac{1}{np'^{\ln\ln(n)}q'^{\ln\ln\ln(n)}}$$ For $p,q\in\mathbb{R}$. Obviously, $...
1
vote
2answers
260 views

How to test for convergence for the harmonic series with irregular (binomial) sign changes? [closed]

How would you test for convergence with a series such as this using the alternating series test? $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}...-\frac{1}{10}+\frac{1}{11}...
-1
votes
3answers
131 views

$f$ such that $\int_1^{\infty}f(x)dx$ converges, but not absolutely? [closed]

What's an easy example of a function $f$ such that $$\int_1^{\infty}f(x)dx$$ converges, but not absolutely?
2
votes
2answers
451 views

Improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $$\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$$ for a constant $C>0$ and conclude that the ...
2
votes
1answer
574 views

Normed space where all absolutely convergent series converge is Banach [duplicate]

Let $A$ be a normed space where every absolutely convergent series converges in $A$. How do I prove that $A$ is Banach? Let $\sum_{n=1}^\infty x_n$ be an absolutely convergent series in $A$, then ...
6
votes
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 ...
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.
5
votes
1answer
607 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 $(\sum x_n)$ ...
7
votes
0answers
103 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 \...
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 ...
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{...
2
votes
1answer
383 views

Exponential of a complex number converges absolutely

$$\operatorname{exp}(z)=\sum_{n=0}^\infty \frac{z^n}{n!}$$ This converges absolutely for every $z\in \Bbb C$. What does this mean to a layman?
2
votes
1answer
24 views

Why is $\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolut convergent?

Why is $\sin(tA)=\sum\limits_{k\ge0}\frac{(-1)^k}{(2k+1)!}t^{2k+1}A^{2k+1}$ absolute convergent ? for a $n\times n$ real matrix $A$ and $t\in \mathbf R$ Which crieterion is to use ?
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
votes
2answers
126 views

Definition for the complex exponential function

We define the exponential function as $\exp(z)=\sum\limits_{j=0}^\infty= \frac{z^j}{j!}$ for all $z\in \mathbb{C}$. Lets now compute $\exp(0)$, then we would have to calculate $0^0$ which undefined. ...
1
vote
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
3answers
51 views

Testing convergence of series $\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$

Lets have a series $$\sum_{n=2}^\infty\frac{\ln\left(\frac{n+1}{n-1}\right)}{\sqrt{n}}$$ However, I have absolutely no clue how to try to continue. I could probably use the integral criterion and ...
1
vote
2answers
178 views

Splitting an infinite series

Let $A$ be a countably infinite set, and let $f:A\to\mathbb{R}$ be a function that takes elements of $A$ to the reals. Suppose that $\sum_{w\in A} f(w)$ is well-defined (see note below). Also suppose $...
1
vote
4answers
148 views

Series and Sequences - Absolute Convergence

I'm practicing for my final exams this week but the past year papers have no answers so I'm not sure if my answers are acceptable, was hoping someone would look at my proof and let me know if it is a ...
1
vote
1answer
78 views

Prob. 12, Chap. 3 in Baby Rudin: Some results involving the remainder of a convergent series of positive term series

Here is Prob. 12, Chap. 3 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Suppose $a_n > 0$ and $\sum a_n$ converges. Put $$ r_n = \sum_{m=n}^\infty a_m.$$ ...
0
votes
1answer
63 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
0
votes
1answer
125 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....
0
votes
0answers
46 views

Does convergence of $\Gamma(x)$ imply convergence of $\Gamma(z)$? Is it generalisable?

If we have the gamma function in integral form $\Gamma(x)=\int_\limits0^\infty e^{-t}t^{x-1}dt$ and have proven that it converges for real $x>0$ (to a point), can we then immediately conclude that $...
0
votes
2answers
177 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
1answer
117 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 $\...