# 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 ...