State whether the below series converges or diverges and justify your response through the adaptive steps.

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n!}{(n+2)!}$

For reference, we will use $a_n=\frac{n!}{(n+2)!}$

First, simply $a_n$ by expanding the factorials and canceling. It is recommended that you preview your answer to ensure the correct use of parentheses. Use a multiplication sign between terms even with parentheses, such as $x^{**}(x+1)$ for $x(x+1).$

Simplified $a_n=$