i guys.

i need to determine if this serie converges:

$\displaystyle \sum_{n=2}^{\infty}\frac{1}{n\ln(n)\ln(\ln(n))}$

this is what i've tried so far:

i use condensation test and got the following:

$\displaystyle \sum \frac{2^n}{2^n\ln2^n[\ln(\ln2^n)]}=\frac{1}{n\ln2[\ln(n\ln2)]}$

$\displaystyle \frac{1}{\ln2}\sum \frac{1}{n[\ln(n\ln2)]}=\frac{1}{n[\ln n+\ln(\ln2)]}=\frac{1}{n\ln n+n\ln(\ln2)}$

$\displaystyle \frac{1}{\ln2}\sum \frac{1}{n\ln n+n\ln(\ln2)}=\frac{1}{n\ln n}+\frac{1}{n\ln(\ln2)}$

now, my question is this:

can i say that since: $\displaystyle \sum \frac{1}{n\ln(\ln2)}$ does not converge the the whole thing does'nt converge?

is there more elegant way to handle this horrible serie?

thanks in advanced!