# Given a divergent series, which alterations are convergent?

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 $\sum \frac{a_n}{1+n^2a_n}$?

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I suspect that the first series always diverges and I know that the second one always converges by comparison to $1/n^2$ –  Mark Feb 19 '13 at 22:48
See here. –  Chris Feb 19 '13 at 23:18