Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

420 questions
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 ...
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 ...
207 views

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 ...
322 views

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. ...
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 ...
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 ...