TITLE: Find a series convergent $\sum a_n$ such that $\sum\sqrt{a_n/n}$ diverges.
QUESTION [5 upvotes]: This is part of an exercise 8.22 from Apostol's Mathematical Analysis.  I've looked at things like, $a_n=1/\sqrt n-1/\sqrt{n+1}$, and $a_n=1/\log n^{\log n}$, but I can't seem to find anything that works.

REPLY [10 votes]: Try $a_n=\frac{1}{n\log^2 n}$.