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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 $\sum_{n=0}^{\infty}\frac{1}{(n+2)}.\frac{x^n}{n!}$

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