TITLE: Prove that if $\sum_{n=1}^ \infty na_n$ converges, then $\sum_{n=1}^ \infty a_n$ converges.
QUESTION [6 upvotes]: Prove that if $\displaystyle \sum_{n=1}^ \infty na_n$ converges, then $\displaystyle\sum_{n=1}^ \infty a_n$ converges. 
No, $a_n$ are not necessarily positive numbers.
I've been trying summation by parts.

REPLY [4 votes]: Dirichlet's Test with {$na_n$} and {$1/n$}?

Since $\sum_{n=1}^\infty na_n$ converges, the partial sums
$\sum_{n=1}^\infty na_n$ are bounded.
$\lim \frac {1}{n} = 0$.
The sequence $\frac {1}{n}$ is monotone, therefore the series $\sum_{n=1}^\infty \left\lvert \frac {1}{n+1} - \frac   
   {1}{n}\right\rvert$ converges and the conditions hold for the test.
So, $\sum_{n=1}^\infty (\frac {1}{n})(na_n)$=$\sum_{n=1}^\infty a_n$ converges.