if $\sum_{n=1}^{\infty }a_{n}$ converges ($a_{n}$ are non negatives), study the convergence of the serie $\sum_{n=1}^{\infty }\sqrt{\frac{a_{n}}{n}}$

Question

Does it converges? Or it depends on $a_{n}$ terms? I would thank to you if you can explain me if there are some cases in which the serie diverges and other in which the serie converges or if it just converges or if it just diverges.

Answer 1

It depends on $a_n$. For example, when $a_n=\frac{1}{n^2}$, it converges; when $a_n=\frac{1}{n(\log n)^2}$ for $n\ge 2$, it diverges.