# Convergence or divergence of a series given divergent series

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 $$0<\sum \frac{a_n}{1+n^2a_n} < \sum \frac{a_n}{n^2a_n} = \sum \frac{1}{n^2}$$
• (+1) Looks like the assumption that $\sum a_n$ diverges is unnecessary. – Yanko Jan 17 at 22:00