If the sum $\sum_{n=1}^\infty n a_n$ converges for positive $a_n$, what can we say about the sequence $a_n$?

Let $(a_n)$ be a sequence of positive numbers and assume that

$$\sum_{n=1}^\infty n a_n < \infty.$$

What can we then say about $a_n$? There are a few obvious things; $a_n\to 0$ and the series $\sum_{n=1}^\infty a_n$ converges. I would like a result along the lines of: For all sufficiently large $n$, $a_n\le b_n$ for some summable sequence $(b_n)$ not depending on $a_n$. I feel like $a_n$ should behave like $1/n^2$ asymptotically, but I have not been able to show anything like that. Any help is appreciated.