Answer
Diverges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n! }{n^9}$$
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)! }{(n+1)^9} \frac{n^9}{n! }\\
&=\lim _{n \rightarrow \infty} \frac{(n+1)n! }{(n+1)^9} \frac{n^9}{n! } \\
&=\lim _{n \rightarrow \infty} \frac{n^9 }{(n+1)^8}\\
&=\infty
\end{align*}
Thus the series diverges.
