If $\displaystyle\sum\limits_{n=0}^{\infty}a_n$ is divergent and $a_n > 0$ for all $n$, then does $\displaystyle\sum\limits_{n=0}^{\infty} \dfrac{a_n}{1+n^2 a_n}$ converge or diverge?
The only progress I have is that if you consider the harmonic series, then we get the series with terms $\dfrac{1}{n(n+1)}$, which converges.
HINT You have $$ 0<\sum \frac{a_n}{1+n^2a_n} < \sum \frac{a_n}{n^2a_n} = \sum \frac{1}{n^2} $$