Honestly, It is a homework problem.
Let $\sum_{n=1}^{\infty} x_n$ be a divergent series with positive terms. We have to examine whether the following are true or false
i) $\sum_{n=1}^{\infty} \frac{x_n}{1+n^2x_n}$ is convergent
ii) $\sum_{n=1}^{\infty} \frac{x_n}{1+nx_n} $ is divergent
I am able to do the first one by comparing with the series $\sum_{n=1}^{\infty} \frac{1}{n^2} $
But the second one I am unable to do.
If $x_n$ $\geq$ $1$ then of course it is divergent. But in general I am unable to do it.
Please help.
I really do not want the full steps.
I just need a hint.
(I have tried by Comparison test or its limiting form)