Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

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4
votes
1answer
111 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 ...
1
vote
2answers
173 views

Series $ \sum_{n = 0}^\infty \frac{(-1)^n\sin(n)}{n!} $ is absolutely convergent?

I'm having trouble proving the series $$ \sum_{n = 0}^\infty \frac{(-1)^n\sin(n)}{n!} $$ is absolutely convergent. My try I know that the series $$ \sum_{n = 0}^\infty \frac{\sin(n)}{n!} $$ ...
3
votes
6answers
175 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 ...
0
votes
2answers
159 views

absolutely convergent series and its properties

Assume we have two absolutely convergent series $\{a_n\}^{\infty}_{n=1}$ and $\{b_n\}^{\infty}_{n=1}$ such that $\sum^{\infty}_{n=1}a_n=\sum^{\infty}_{n=1}b_n$ and $\lim_{n\to\infty}|{\frac{a_n}{b_n}}|...
1
vote
1answer
234 views

If $\sum_{n=1}^\infty |a_n|^2<\infty$, Then : $\sum_{n=1}^\infty a_n$ Converges $\Leftrightarrow \prod_{n=1}^\infty(1+a_n)$Converges

Prove that for complex sequence $\{a_n\}_{n\in\mathbb{N}}$ : if $\displaystyle \sum_{n=1}^\infty |a_n|^2<\infty$, Then : $\displaystyle \sum_{n=1}^\infty a_n$ Converges $\Leftrightarrow \prod_{...
1
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2answers
121 views

Alternating series variant of a convergent series

I'm trying to prove whether the following series is convergent, divergent, or that there is not enough info. If the series ∑ an is convergent and has positive terms, what is the series below? $$ \...
0
votes
0answers
161 views

Does Mertens convergence theorem say that the resulting series is absolutely convergent?

Here's the wikipedia article on Convergence and Mertens Theorem. It doesn't say whether the resulting sequence which is the Cauchy product of two absolutely converging sequences is itself absolutely ...
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
1answer
570 views

Finding Region of Convergence

I am trying to find the regions of absolute and uniform convergence for three different series, but I figured I'd start with the simplest one $$\sum_{k\geq 2} k(k-1)z^{k-2}$$ I have worked with ...
1
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2answers
69 views

Check: Convergence of an infinite series

More a check than a question - I just need to ensure that my logic is correct (I always had trouble with this): Show whether the series $$\sum_{n=110}^{\infty}\frac{1}{3^{n}n^{3}}$$ Is divergent, ...
3
votes
1answer
91 views

Determine if the series $\sum\nolimits_{n = 1}^{+\infty} \frac{(-1)^n ((n - 1)!!)^2}{n!}$ converges absolutely, conditionally, or diverges

As per the title, I am trying to find if the series $$ \sum\limits_{n = 1}^{+\infty} \frac{(-1)^n ((n - 1)!!)^2}{n!} $$ converges absolutely, conditionally, or diverges. The $“!!”$ is the double ...
0
votes
4answers
1k views

Testing for absolute convergence? $\frac{(-1)^n}{5n+1}$

I'm trying to test the summation $\sum^\infty_{n=0}\frac{(-1)^n}{5n+1}$ for absolute convergence. By alternating series test, I can tell is is at least conditionally convergent. However, when I used ...
2
votes
2answers
103 views

Showing power series converges absolutely

Show that if the sequence ${a_n}$ is bounded then the power series $\sum a_nx^n$ converges absolutely for $|x|<1$. I haven't the slightest idea how to prove this. Does anyone have any thoughts on ...
0
votes
2answers
379 views

Power Series — Convergence, Divergence, and Absolute Convergence

Suppose that the power series $$\sum a_nx^n$$ is convergent at $x=-3$ and divergent at $x=5$. What can be said about the following: convergence at $x=-2$ ? absolute convergence at $x=2$ ? ...
2
votes
3answers
1k views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
11
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2answers
210 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
2answers
511 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 ...
1
vote
2answers
40 views

Connection between series

I have to show that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} a_n^2$ is absolutely convergent too. Please give me some hint, how do I start the excercise. ...
3
votes
2answers
269 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 ...
0
votes
2answers
229 views

Convergence or divergence of series where the terms include reciprocal of the natural logarithm

Show whether the following series - $ \sum_1^{\infty}\frac{1}{n\ln(n)} $ converges or diverges. Is it possible to make in particular a clever use of the limit comparison test (described in the ...
0
votes
1answer
105 views

A different type of Convergence of Fourier Series

I have just started studying fourier series. All the convergences I have seen considered the partial sums to be $\sum\limits_{i=-n}^n a_n Sin(n\theta)$. But in all practical systems the harmonics ...
1
vote
1answer
128 views

Find an absotule convergent series that is not convergent

find the sequence of polynomials $(P_n)$ such that $\sum P_n$ converges absolutely (that is $\sum \|P_n\|_{\infty}\lt\infty $) but is not convergent in the space ($\mathcal{P}[0,1], \|.\|_{\infty}$, i....
2
votes
1answer
82 views

Convergence of $\sum_{k=1}^{\infty} \frac{2k+1}{10k+2} \sin(\frac{k\pi}{2}) $

Task is to determine if the infinite sum converges absolutely, conditionally or doesn't converge. $$\sum_{k=1}^{\infty} \frac{2k+1}{10k+2} \sin\left(\frac{k\pi}{2}\right)$$ I have determined that ...
5
votes
3answers
255 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 $\sum_{n \geq 1} n a_n^2 < \infty$ ...
1
vote
1answer
86 views

Does $\sum_{n=0}^\infty\frac{a^n}{\frac{n}{2}!}x^n$ converge?

And if so, what is the radius of convergence of $x$? I am inclined to think it converges absolutely for all $x$ but I can't prove it. I have tried using an adaptation of the ratio test: $\rho=\lim\...
2
votes
0answers
113 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
vote
1answer
136 views

Find the converging sum of a series

I've had a test in probability, during which I've had to find the convergence value of $\sum\limits_{i=1}^\infty n^2 \dot (x)^{n-1} $ when $x=\frac{5}{6}$. I couldn't find the answer during the ...
0
votes
1answer
561 views

absolutely convergent series in Hilbert space

Is it possible to find an infinite dimensional Hilbert space, where every convergent series is absolutely convergent? I could not find any clue to find an example of such type or to disprove that. ...
0
votes
1answer
1k views

how to prove the convergence of fixed point iteration algorithm

Please refer to the below algorithm: Above two steps can be rewritten as, \begin{equation} x(k+1)=\arccos\bigg( -\frac{1}{2(Dr^{\frac {|\sin(2x(k)+\theta)|}{M\sin x(k)\sqrt{A+2B\cos(2x(k)+\theta)}}}+...
1
vote
1answer
39 views

On (absolute) convergence of $f_c:= c + \sum_{n=0}^{+ \infty} \frac{a_n}{n+1}x^{n+1}$

Let $R> 0$ and let $g: (-R,R) \longrightarrow \mathbb{R}$ be given by the convergent power series $$g(x):= \sum_{n=0}^{+ \infty} a_nx^n$$ for $|x| < R$. Let $c \in \mathbb{R}$ and let $f_c: (-R,...
0
votes
1answer
159 views

Determining $a$ values for convergence in alternating series

So we have this series: $$ \sum^{\infty}_{n = 1}\left(-1\right)^n\sin\left(a \over n\right) $$ The task is to find all $a$ values, in which case the series converges and all $a$ values in which case ...
2
votes
1answer
355 views

Show absolute and uniform convergence of a Fourier series

Hello and good evening! The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ I have to examine if ...
0
votes
1answer
333 views

Absolute convergence of a double sum.

If $A\in\mathbb{C}^{\infty\times \infty}$ such that $$ \sum_{i=0}^\infty \left|\sum_{j=0}^\infty a_{ij}\right|<\infty,\qquad \sum_{j=0}^\infty \left|\sum_{i=0}^\infty a_{ij}\right|<\infty,$$ ...
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 ...
2
votes
3answers
198 views

If $\sum c_n$ converges absolutely and $k_n$ is bounded, does $\sum c_nk_n$ converge absolutely?

Suppose a series $\sum\limits c_n$ converges absolutely and a sequence $k_n$ is bounded. Will the sequence $\sum\limits c_nk_n$ converge absolutely? Since $k_n$ is bounded there must exist an ...
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 $\sum _{n=1}^\infty a_n$ ...
16
votes
1answer
6k 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
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, ...