How do I find all the different cases of convergence/divergence of this series?

Question

Find if sum converges or diverges \begin{equation} \sum_{n=1}^{\infty}\frac{ 1}{n(\ln n)^p(\ln(\ln n)^q } \end{equation} I know how to answer the question if $q=1$ and $p=1$ (through integral comparison) But I'm can't seem to solve other cases. Any tips?

Answer 1

Hint:

If $p>1$, the Bertrand's series $\; \displaystyle\sum_{n\ge 2}\frac1{n\ln^p n}$ converges (by the integral test) and $$\frac{ 1}{n\ln^pn \,\bigl(\ln(\ln n)\bigr)^q}=o\biggl(\frac{ 1}{n\ln^pn}\biggr)\dots $$