Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{(\ln n)^n}{n^{n}}$
In order to solve the given series we will take the help of Root Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges.
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{(\ln n)^n}{n^{n}}|}$
$\implies \lim\limits_{n \to \infty}\dfrac{\ln n}{n}=\dfrac{\infty}{\infty}$
Apply L-Hospital's rule to find the indeterminate form of the limit.
$L=\lim\limits_{n \to \infty}\dfrac{\dfrac{1}{n}}{1}=0 \lt 1$
Hence, the series converges by the Root Test.