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 $\displaystyle \sum_{n\ge1} e^{2n\pi i t}a_n$ converges for all $0\le t< 1$.
What about the converse implication? That is,
- Assume $\displaystyle \sum_{n\ge1} e^{2n\pi i t}a_n$ converges for all $0\le t< 1$. Does this imply $\displaystyle \sum_{n\ge 1} |a_n|<+\infty$?