Convergence of $\sum_1^\infty \ln (\frac{3+n^p}{2+n^p})$ I am able to prove divergence for $p<0$ or $p=0$.
How can I prove convergence/divergence for $p>0$.

You got:
$$\ln \left( \frac{3+n^p}{2+n^p}\right) = \ln \left( 1+ \frac{1}{2+n^p}\right)$$
hence the well known asymptotics $\ln (1+y)\approx y$ as $y\to 0$ and $p>0$ yield:
$$\ln \left( \frac{3+n^p}{2+n^p}\right) \approx \frac{1}{2+n^p} \approx \frac{1}{n^p}\; ;$$
therefore your series converges if and only if $p>1$.
