Answer
Converges
Work Step by Step
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. In order to solve the series we will take the help of Root Test.
It can be defined as follows: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
Let us consider $a_n=\dfrac{1}{n^{(1+n)}}$
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{1}{n^{1+n}}|}$
$\implies \lim\limits_{n \to \infty}\dfrac{1}{(n)^{(1/n)}(n^n)^{(1/n)}}=\dfrac{1}{\infty}=0 \lt 1$
Hence, the series Converges by the Root Test.