Showing convergence of an infinite series?

I'm trying to show whether the below series converges or diverges, and I have very little clue on how to do it. I know about the Comparison Test, but I can't think of a sequence $b_n > a_n$ to perform a comparison with.

Esentially, my question is more how should I approach this problem, and is there a less "guess-based" approach for finding a $b_n$?
$$\sum_{n=2}^\infty a_n =\sum_{n=2}^\infty\frac{1}{\log(\log(n))\log(n)n}$$