Firstly we are given that $a_k\in\mathbb C$ and $\sum\limits_{k=1}^\infty {a_k}^2$ converges absolutely. The question then asks to show that $\sum\limits_{k=1}^\infty \frac{a_k}{k}$ converges absolutely.
I know that absolute convergence of the first series this implies that $\sum\limits_{k=1}^\infty ||{a_k}^2||$ converges.
Thus by the Cauchy-Schwarz inequality I get
$$
\sum\limits_{k=1}^\infty ||{a_k}^2|| \leq \sum\limits_{k=1}^\infty ||{a_k}||.||a_k||
$$
From this point onwards I have no idea of how to proceed. (Should I make use of Cauchy products?)
Best Answer
$$\sum_{k=1}^{\infty}\frac{a_k}{k} \leq \sqrt{\left(\sum_{k=1}^{\infty}a_k^2\right ) \left ( \sum_{k=1}^{\infty}\frac{1}{k^2} \right ) }$$
By Cauchy-Schwarz.