Answer
Diverges
Work Step by Step
Given
$$\sum_{n=0}^{\infty} \frac{n! }{6^n}$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{(n+1)! }{6^{n+1}} \frac{6^n}{n! }\\
&= \lim _{n \rightarrow \infty} \frac{n+1}{6} \\
&=\infty
\end{align*}
Thus the series diverges.
