The given series is \(\sum_{n=1}^\infty\frac{(2n)!}{n^5}\)

Compare the above series with its standard form \(\sum_{n=1}^\infty a_n\) and obtain that \(a_n=\frac{(2n)!}{n^5}\)

Apply the ratio test as follows.

\(\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{(2n+2)!}{(n+1)^5}\times\frac{n^5}{(2n)!}\)

\(\lim_{n\to\infty}\frac{(2n+1)(2n+2)}{(1+\frac{1}{n})^5}\)

\(=\infty\)

\(>1\)

By ratio test,

the series \(\sum_{n=1}^\infty\frac{(2n)!}{n^5}\) diverges.

Compare the above series with its standard form \(\sum_{n=1}^\infty a_n\) and obtain that \(a_n=\frac{(2n)!}{n^5}\)

Apply the ratio test as follows.

\(\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{(2n+2)!}{(n+1)^5}\times\frac{n^5}{(2n)!}\)

\(\lim_{n\to\infty}\frac{(2n+1)(2n+2)}{(1+\frac{1}{n})^5}\)

\(=\infty\)

\(>1\)

By ratio test,

the series \(\sum_{n=1}^\infty\frac{(2n)!}{n^5}\) diverges.