Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{(\ln n)^n}{n^{n}}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{(\ln n)^n}{n^{n}}|}=\lim\limits_{n \to \infty}\dfrac{\ln n}{n}$
or, $=\dfrac{\infty}{\infty}$
This shows an indeterminate form of a limit so, apply L'Hospital's rule.
So, $\lim\limits_{n \to \infty}\dfrac{1/n}{1}=0 \lt 1$
Thus, the given series Converges by the Root Test.