Answer
The series is divergent.
Work Step by Step
Let $a_{n}=\frac{1}{n^{1+1/n}}$ and $b_{n}=\frac{1}{n}$
$\lim\limits_{n \to \infty}\frac{a_{n}}{b_{n}}=\lim\limits_{n \to \infty} \frac{1}{n^{1/n}}$
$=\lim\limits_{n \to \infty} \frac{1}{e^{lnn/n}}$
$=1$
The series is divergent.