I basically need help with the algebra involved in this problem. Specifically an explanation of how the ! and the 2^n are going to cancel out. Thank You.
ratio test ...
$\displaystyle \displaystyle \lim_{n \to \infty} \frac{(n+2)!}{2^{n+1}} \cdot \frac{2^n}{(n+1)!}$
note that $\displaystyle \displaystyle \frac{(n+2)!}{(n+1)!} = n+2$
and the limit simplifies to ...
$\displaystyle \displaystyle \lim_{n \to \infty} \frac{n+2}{2}$