I have to show that $\displaystyle \sum\_{n=2}^{\infty} \frac{\cos(\pi n)}{\log(\log(n))}$ isn't absolutely convergent.

I thought about doing the following:

$\begin{align}\displaystyle\sum\_{n=2}^{\infty}\left|\frac{\cos(\pi n)}{\log(\log(n))}\right|&=\sum\_{n=2}^{\infty} \frac{1}{|\log(\log(n))|} \geq \sum\_{n=2}^{\infty} \frac{1}{|\log(n-1)|} \geq\\&\geq\sum\_{n=2}^{\infty} \frac{1}{n}.\end{align}$

My problem is, that I don't know if

$\big|\log(\log(n))\big|\leq\big|\log(n-1)\big|\leq n$

is a true statement. Can anybody help me out?