How would I express $$\sum_{k=1}^{n}\frac{1}{(k+2)k!}$$ in terms of $n$?
An attempt of mine is $$\sum_{k=1}^{n}\frac{1}{(k+2)k!} = \sum_{k=1}^{n}\frac{1}{(k+1)! + k!},$$ which is not useful for me. Is there any established expression for a sum involving factorials in terms of $n$?
An elementary approach is of the first priority, say high-school math.
$$\frac{1}{(k+2)k!} = \frac{k+1}{(k+2)!} = \frac{1}{(k+1)!} - \frac{1}{(k+2)!}.$$