calculus - series comparison test for $1/{n^{p}log(n)}$

It is really not obvious to me which values of $p$ make $\sum_2^\infty \frac{1}{n^{p}log(n)}$ converge only using comparison test (general comparison test, limit comparison test).

Am I missing something ? I knew that the series converges for $p > 1$ and diverges for $p \leq 0$ by comparing it with $1/n^{p}$ but that's it.

Edit. I know it can be solved using integral test, cauchy condensation test, etc. but I am just wondering if there is a comparison to some other series that directly results in divergence of this one for $p \leq 1$ (or just $p < 1$ is fine, too.)

Answer

For $p<1$ compare with $\sum 1/n^q$ with $p0$\lim_{n\to\infty}\frac{\log n}{n^\epsilon}=0.$$