All questions with tag [math: absolute-convergence]

1

$b_n$ bounded, $\sum a_n$ converges absolutely, then $\sum a_nb_n$ also

a) Prove that if $\sum a_n$ merges definitely and also $b_n$ is a bounded series, after that additionally $\sum a_nb_n$ merges definitely. I intended to make use of the contrast examination to show it holds true, yet I assume I obtained something blended. Below is what I have until now. $b_n$ is bounded $\Rightarrow \exists c > 0: |b_k|...
2022-06-29 22:04:02
3

Does absolute convergence of a sum imply uniform convergence?

Suppose I have a collection $\sum_{n = 0}^{\infty} f_{n}(x)$ which merges definitely to a function $f(x)$. Does the collection merge evenly to $f(x)$? I intend to claim this adheres to from Dini's Theorem, yet I can not appear to see just how.
2022-06-27 14:08:00