How to find a compact expression for $\sum\limits_{k=1}^n \frac k{(K+1)!}$?

Question

$$\sum_{k=1}^n \frac k{(K+1)!}$$

How to find a compact expression?

(Original scan here)

Answer 1

HINT:

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

Can you recognize the Telescoping Series?