Prove that if $\sum |a_n|^2$ converges then $ \sum \frac{a_n}{n}$ converges Let $ \{a_n\}\in \mathbb{C}$. Prove that if $\sum |a_n|^2$ converges than $\sum \frac{a_n}{n}$ converges.
Note that this problem is taken from the first chapter on series of a calculus book, so it should be solvable with very basic tools (e.g. the only convergence test introduced was the comparison test and nothing about absolute convergence is assumed).
I have tried a few thing but nothing worked. Could you give me a hint on how to approach the problem?

With Cauchy-Schwarz inequality and the fact that $\sum_n 1/n^2 < \infty$, the result follows.
