How to tell which values of $p$ let this sum converge

Question

This is the sum: $$\sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p}$$ How do I tell which values of $p$ allow this to converge? The ratio test isn’t working out for me at all.

Answer 1

The ratio test will not work because this series converges/diverges far too slowly for the test to give any information. Rather, the integral test is most useful when studying things related to $p$-series. In particular, if we set $u = \ln \ln x$ then we can find that

\begin{align*} \int_3^{\infty} \frac{1}{x \ln x (\ln \ln x)^p} \, dx &= \int_{\ln \ln 3}^{\infty} \frac{1}{u^p} \, du \end{align*}

which is now a vastly easier integral to study.