If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges

Question

Given $a_n>0$.
If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges?

My approach:

Ihad used atmost every method like 1)AM and GM inequality
2) cauchy Swartz inequality
3) by definition of convergent series and so on..
I am not able to prove this statement.
I am thinking now this statement is false, neither I am able to find counter example for this.
Any Hint/direction please..

Answer 1

Counterexample: $a_n=1/k$ if $n=k^2$ with $k \in \mathbf{N}$, and $a_n=0$ otherwise. In this case, $\sqrt{a_n/n} = 1/k^{3/2}$ if $n=k^2$ with $k \in \mathbf{N}$, and $\sqrt{a_n/n} = 0$ otherwise.