Use the Ratio Test $to$ determine whether the series $is$ convergent $or$ divergent.

${\displaystyle \underset{n=1}{\overset{}{}}}(-1{)}^n\frac{5^{n}n!}{10*17*24*cdots*(7n+3)}$

Identify $|a_n|.$

$\frac{5^{n}n!}{10*17*24*cdots*(7n+3)}$

$\frac{5^n}{(7n+3)!}$

$\frac{5^{n}a!}{(7n+3)!}$

$\frac{(5^{n}n)!}{10*17*24*cdots*(7n+3)}$

$\frac{5^{n}n}{(7n+3)!}$

Evaluate the following $limit.$

$\underset{n}{lim}|\frac{a_{n+1}}{a_n}|$

Since $,$