Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{n}{(\ln n)^n}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{n}{(\ln n)^n}|)^{1/n}=\lim\limits_{n \to \infty} \dfrac{\sqrt [n] n}{ \ln n}$
So, $l=\dfrac{1}{\infty}=0 \lt 1$
Thus, the given series Converges by the Root Test.