Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{(n!)^3 }{(3n)!}$$
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)!)^3 }{(3n+3)!} \frac{(3n)!}{(n!)^3 }\\
&=\lim _{n \rightarrow \infty} \frac{((n+1)n!)^3 }{(3n+3)(3n+2)(3n+1)(3n)!} \frac{(3n)!}{(n!)^3 } \\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^3 }{(3n+3)(3n+2)(3n+1)} \\
&=\frac{1}{27}<1
\end{align*}
Thus the series converges.
