Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{1}{n^{n}}$$
By using the Ratio Test
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(n)^{n}}{(n+1)^{n+1}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{(n)^{n}}{(n+1)(n+1)^{n}}\\
&=\lim _{n \rightarrow \infty} \frac{(n)^{n}}{ (n+1)^{n}}\lim _{n \rightarrow \infty}\frac{1}{n+1}\\
&=0<1
\end{align*}
Thus the series converges.
