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 $$ 0<\sum \frac{a_n}{1+n^2a_n} < \sum \frac{a_n}{n^2a_n} = \sum \frac{1}{n^2} $$

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    $\begingroup$ (+1) Looks like the assumption that $\sum a_n$ diverges is unnecessary. $\endgroup$ – Yanko Jan 17 at 22:00
  • $\begingroup$ @Yanko likely professor tried to trick them into thinking this may diverge as well, not sure. Definitely not needed... $\endgroup$ – gt6989b Jan 17 at 22:00

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