Answer
The series is absolutely convergent by root test.
Work Step by Step
$\sum_{n=1}^{\infty}a_{n}=\sum_{n=1}^{\infty}\frac{(-2)^n}{n^n}$
$|a_{n}|=\frac{2^n}{n^n}=(\frac{2}{n})^{n}$
$\lim\limits_{n \to \infty} \sqrt[n] |a_{n}|=\lim\limits_{n \to \infty} \sqrt[n] {(\frac{2}{n})^{n}}$
$=\lim\limits_{n \to \infty} {\frac{2}{n}}$
$=0\lt 1$
The series is absolutely convergent by root test.
