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