TITLE: Another simple series convergence question: $\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$
QUESTION [5 upvotes]: I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the convergence of $\displaystyle\sum\limits_{n=3}^\infty\frac 1{n\ln (n\ln 2)}$. Am I on the right path, and how do I proceed from here?

REPLY [5 votes]: Apply directly integral test 
$$\int_3^{\infty}\frac{(\ln(\ln x))'}{ \ln(\ln x)}\mathrm{dx}= \left[\ln(\ln(\ln(x)))\right]_3^{\infty}\longrightarrow \infty$$