# Questions tagged [absolute-convergence]

This tag for questions related to absolute convergence a series.

438 questions
Filter by
Sorted by
Tagged with
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 ...
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 $$\sum_{n = 0}^\infty \frac{\sin(n)}{n!}$$ ...
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 ...
159 views

121 views

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 ...
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 ...
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 $\sum _{n=1}^\infty a_n$ ...
### $b_n$ 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 ...
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, ...