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:

1. First, I tried separating it into two cases: $a_n \to \infty$, $a_n \to A$ (where $A$ is some const value).

2. 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.

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Those are not the only cases. a_n can oscillate or even tend to zero slowly. (Casework is the way to go, though.) – Qiaochu Yuan Nov 29 '10 at 23:04

Hint: Rewrite as $$\sum_{n=1}^{\infty} \frac{1}{\frac{1}{a_n} + n^2}.$$
I was thinking about it, but I was not sure I can divide it by $a_n$. Thanks for the clarification. – Ma.H Nov 29 '10 at 22:56
@Ma. H: Maybe I misunderstood. I read your first sentence as implying that all the terms in the series are positive and that the series diverges. Is that not correct? If $a_n >0$ for all $n$ then you can divide by $a_n$. – Mike Spivey Nov 29 '10 at 23:08
From this it is clear that the series converges for any positive sequence $a_n$, so the assumption that $\sum a_n$ diverges is irrelevant. – Hans Lundmark Nov 30 '10 at 6:32
Divide the numerator and denominator by $a_n$, and compare what you get with $1/n^2$.