Does $\sum_{n=1}^\infty a_n^2 \text{ converges },\ a_n \geq 0 \implies \sum_{n=1}^\infty \frac{a_n}{n} \text{ converges}$?
If $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$