Answer
Converges
Work Step by Step
Apply the ratio test to the series
$\Sigma^{\infty}_{n=1} \frac{1}{n^n} $
$\lim\limits_{n \to \infty} \sqrt[n] {|\frac{1}{n^n}| }$, Absolute values fall out because all values are positive.
$\lim\limits_{n \to \infty} \frac{1}{n} = 0 <1$
Because the answer is less than 1, the series converges
