Problem: The series $\sum_{n=1}^{\infty} a_n$ diverge and positive. What can be said on the series $\sum_{n=1}^{\infty} \frac{a_n}{1+n^{2}a_n}$ ?
Approach:
First, I tried separating it into two cases: $a_n \to \infty$, $a_n \to A$ (where $A$ is some const value).
Tried the ratio test and got nowhere with that.
I think that the right approach is to check each case separately, but my hunch tells me that there is a workaround.