Convergence

Hi,

I have the following series: \sum_{n=2}^{\infty}\frac{1}{n\,\textup{log}^{2}n}

I just want to know whether it converges or not. I've tried all the convergence tests I've found (at least inf limit, ratio, root, Raabe, Cauchy, maybe some other I forgot?), none is conclusive. I've tried to numerically sum it (just to get an idea), but with floating point doubles I run out of precision and with arbitrary precision arithmetic I run out of patience before reaching any conclusion.

My hunch it that it doesn't converge. It's not really important but it's been bugging me, I must have missed something obvious.

Let me try that again: \sum_{n=2}^{\infty}\frac{1}{n\,log^{2}n}

What's the base of your logarithm?

>>3
Irrelevant, that would just scale the series by a constant.

Integral test, substitute log n for t.

It converges.

>>4
Irrelevant, that would just scale the series by a constant.
Nonsense.

>>5
blatherskite.

:/

>>5
I don't understand.

http://upload.wikimedia.org/math/5/6/f/56f7df21ad6bc1d3ce9ddff7ac5bb047.png