Find a closed form for the exponential generating function for the sequence $\{a_n\}$ where
$a_n=\frac{1}{(n+1)(n+2)}$
I broke it down into partial fractions and got
$a_n=\frac{1}{n+1}-\frac{1}{n+2}$
We have to give closed form of $\sum_{n =0}^{\infty}(\frac{1}{(n+1)}-\frac{1}{(n+2)})\frac{x^n}{n!}$
I solved for $\sum_{n=0}^{\infty}\frac{1}{n+1}.\frac{x^n}{n!}=\frac{e^x-1}{x}$
Please help me with the part $\sum_{n=0}^{\infty}\frac{1}{(n+2)}.\frac{x^n}{n!}$