Answer
Convergent
Work Step by Step
By Root Test, we have
$\lim_{n\to\infty}\sqrt[n]{\biggl|\frac{1}{(\ln n)^n}\biggr|}=\lim_{n\to\infty}\biggl|\frac{1}{(\ln n)}\biggr|=0$
Since the limit $ < 1$, the series is absolutely convergent therefore convergent