Answer
diverges for all $x$
Work Step by Step
Given $$\sum_{n=0}^{\infty} n^{n} x^{n}$$
Since $a_n = n^{n} x^{n}$ and $a_{n+1} = (n+1)^{n+1} x^{n+1}$, then
\begin{aligned}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(n+1)^{n+1} x^{n+1}}{n^{n} x^{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|x \left(\frac{n+1}{n}\right)^n (n+1 )\right|\\
&=\lim _{n \rightarrow \infty}\left|x \left(1+\frac{1}{n}\right)^n (n+1 )\right|\\
&=|x|\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n\lim _{n \rightarrow \infty} (n+1 ) \\
&=\infty
\end{aligned}
Then the series diverges for all $x$
