calculus - Does the series converge: $sum_2^inftyfrac{log n}{n(log log n)^2}$

Does the series $$\sum_2^\infty\frac{\log n}{n(\log \log n)^2}$$
converge?

The root and ratio tests are inconclusive, the integral test may be too difficult to apply. I've tried the limit comparison test with $\log n/n$ but that is also inconclusive since $$\frac{\frac{\log n}{n(\log \log n)^2}}{\frac{\log n}{n}}=\frac{1}{(\log \log n)^2}\to 0.$$

Answer

Using the Cauchy condensation test, (cf Wikipedia), your series diverges.
The transformed series reads :
$$\sum_{n} 2^n \frac{n}{2^n \log^2 n} = \sum_{n}\frac{n}{\log^2 n}$$

Remark : I interpreted $\log$ as a base two logarithm, which does not affect the result