Determine the value of the series $\sum _{ n=1 }^\infty \frac1{ ( n+2) n!}$

Question

Find the partial sum $S_n$ of the telescoping series

$$\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ \left( n+2 \right) n! } } $$

Answer 1

HINT:

$$\frac1{(n+2)\cdot n!}=\frac{n+1}{(n+2)!}=\frac{n+2-1}{(n+2)!}=\frac1{(n+1)!}-\frac1{(n+2)!}$$