TITLE: Suppose $\sum{a_n}$ converges. How do I prove that $\sum{\frac{\sqrt{a_n}}{n}}$ converges
QUESTION [5 upvotes]: Suppose $\sum{a_n}$ converges. How do I prove that $\sum{\frac{\sqrt{a_n}}{n}}$ converges.
So I know that just because  $\sum{a_n}$ converges, do not mean I can say anything about the converges of its square root. So I know that I can prove that if  $\sum{\sqrt{a_n}}$ converges then $\sum{\frac{\sqrt{a_n}}{n}}$ converges by Abel's Test, but I do not know where to start for the case where  $\sum{\frac{\sqrt{a_n}}{n}}$ diverges

REPLY [10 votes]: Using the AM-GM inequality, $\;\;\displaystyle\frac{\sqrt{a_n}}{n}\le\frac{1}{2}\left(a_n+\frac{1}{n^2}\right)$, so the result follows from the Comparison Test.