TITLE: Convergence of the series $\sum \sqrt{a_n}/n$ given that $\sum a_n$ converges
QUESTION [9 upvotes]: I've been dealing with this problem for enough time and I don't even have a good idea nor a starting point. This is the problem: 

Let $\sum_{n=1}^\infty a_n$ be a convergent series. Show that $\sum_{n=1}^\infty \frac{\sqrt{a_n}}{n}$ is also convergent. 

Thanks for your kind and nice answers, I'm sure a good tip or starting point will suffice.

REPLY [11 votes]: Hint: $\dfrac{\sqrt{a_n}}{n} \leq \dfrac{1}{2}\cdot \left(a_n + \dfrac{1}{n^2}\right)$