For a sequence $a_n \gt 0$ I'm given that $\sum a_n$ diverges.
Let $s_n = a_1 + a_2 + ... + a_n$
It's not immediately obvious, but $\sum \frac{a_n}{s_n^2}$ converges, while $\sum \frac{a_n}{s_n}$ does not.
Knowing these facts, however, can I somehow show divergence or convergence of the series:
$\sum \frac{a_n}{1+na_n}$ and $\sum \frac{a_n}{1+n^2a_n}$?