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> Let $\sum\_{n=1}^\infty a\_n^2$ a converging series with $a\_n \geq 0$. Decide whether $\sum\_{n=1}^\infty \frac{a\_n}{n}$ is converging.
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> Note: you may consider the fact that $\sum\_{n=1}^\infty \frac{1}{n^2}$ is converging.
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I'm having problems with the above task. Given the hint in the note, I believe the latter series is indeed converging and am trying to prove it. Unfortunately, I fail to find a promising approach to do so.

Obviously, $\sum\_{n=1}^\infty a\_n^2 \neq \left(\sum\_{n=1}^\infty a\_n\right) \cdot \left(\sum\_{n=1}^\infty a\_n \right)$, which would be a rookie mistake. Similarly, $\sqrt{\sum\_{n=1}^\infty a\_n^2} \neq \sum\_{n=1}^\infty a\_n$. Therefore, the square seems difficult to get rid of. I thought of arguing using the Cauchy product, however, I'm not sure whether it could be applied "in reverse" to show that if the series based on $a\_n^2$ is converging, then the one based on $a\_n$ is, too.

What approach would you take to solve this problem?