Does this series converge or diverge?
Summation of 2/[k(lnk)^2]
That's a typical stuation where you want to apply Cauchy's Condensation Test (Cauchy condensation test - Wikipedia, the free encyclopedia)
This theorem says that if the terms are decreasing, the series converges or diverges if and only if the series converges or diverges, respectively (i.e. these two series behave the same way as regards convergence/divergence).
So let's take a closer look, therefore, at the series
which turns out to be essentially the same as the well known divergent series
Thus, the series diverges.
Edited:
Plato is right, of course (he is almost always right): I made a silly mistake, since n comes out squared, one gets a series that converges:
So, since converges, the original series converges.