If $\sum\limits_{n=1}^\infty a_n $ is convergent with $a_n$ positive, then $\sum\limits_{n=1}^\infty \frac {{(a_n)}^{1/2}}{n}$ is also convergent

Question

Question: $$\sum_{n=1}^\infty a_n $$ is convergent with $a_n$ positive, prove the series $$\sum_{n=1}^\infty \frac {{(a_n)}^{1/2}}{n}$$ is convergent.

The hint given is: $x^2+y^2\geq2xy$.

But I cannot realize anything from the hint, so asking here for help. I know it has something to do with comparison theorem. Anyone can help ? thanks!

If there is another method without using this hint, please feel free to tell me also! thanks! And also please provide the general method for solving this kind of question if there is any. Thanks so much!

Answer 1

Take the hint with $x = a_n^{1/2}$ and $y = 1/n$.

Answer 2

You can either use $\sqrt{ab}\leq\frac{a+b}{2}$ to state: $$ \sum_{n\geq 1}\sqrt{\frac{a_n}{n^2}}\leq \frac{1}{2}\left(\sum_{n\geq 1}a_n+\sum_{n\geq 1}\frac{1}{n^2}\right)\tag{1}$$ or the Cauchy-Schwarz inequality that gives: $$ \sum_{n\geq 1}\sqrt{\frac{a_n}{n^2}}\leq \sqrt{\frac{\pi^2}{6}\sum_{n\geq 1}a_n}\tag{2}$$ since: $$ \sum_{n\geq 1}\frac{1}{n^2}=\zeta(2)=\frac{\pi^2}{6}.\tag{3}$$