A telescoping sum with factorials

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

Telescoping.
$$S_n=\sum_{k=1}^n\left(\frac1{k!}-\frac1{(k+1)!}\right)=1-\frac1{(n+1)!}\longrightarrow1.$$