Does $\sum^{\infty}_{n=2}\frac{n\pi}{(\ln (n))^2}$ converge or diverge? I need on help on calculus homework.  
$$\sum^{\infty}_{n=2}\frac{n\pi}{(\ln (n))^2}$$
I tried the comparison test, comparing it with:
$$\sum^{\infty}_{n=2}\frac{1}{(\ln(n))^2}$$
but it's not getting me anywhere.
Appreciated for any help!

Hint: The terms do not have limit $0$, so we cannot have convergence. Recall that if $\sum_m^\infty a_k$ exists, then $\lim_{k\to\infty} a_k=0$. (The converse fails.)
Remark: One can think of the above as a special case of Comparison. The series $1+1+1+\cdots$ obviously diverges. It is not hard to show that if $n$ is large enough (and it doesn't really have to be large) we have $\frac{ n\pi}{(\ln n)^2}\gt 1$. 
