TITLE: convergence of $\sum_{n=1}^\infty \sqrt{n}a_n$ implies the convergence of $\sum_{n=1}^\infty a_n$
QUESTION [5 upvotes]: How the convergence  of $\sum_{n=1}^\infty \sqrt{n}a_n$ implies the convergence of $\sum_{n=1}^\infty a_n$? 
I know this if $a_n>0$. But for arbitrary $a_n$, I have no idea...

REPLY [3 votes]: It seems a situation for Abel's test, quite useful in this non absolutely convergent series case: 

The series $c_n = \sqrt{n}a_n $ yields a convergent sum,
$b_n = \frac{1}{\sqrt{n}}$ is monotone and bounded.

Then $\sum_\limits{n} b_n c_n = \sum_\limits{n} a_n $ is convergent too.