Answer
The series converges.
Work Step by Step
We compare the given series
$\sum_{n=1}^{\infty} \frac{1}{n^n} $
with the geometric series
$\sum_{n=1}^{\infty} \frac{1}{2^{n}} $
which is a convergent series ($r=\frac{1}{2}\lt 1$).
Thus, by the comparison test, the given series $\sum_{n=1}^{\infty} \frac{1}{n^n} $ also converges.
