I'm trying to convert $\sum_{k=1}^{n}\frac{k}{(k+1)!}$ into a simple expression. Wolfram Alpha reveals this expression is $1 - \frac{1}{(k+1)!}$ but I unfortunately can't resolve the series of steps taken to reach it. After some research I figured that I probably have to use the telescoping series method in order to cancel out terms so that only a constant number of them remain but I don't see how I can split up this term into a difference for that to work. The only step I can think of is converting the summed term to $\frac{1}{(k-1)!(k+1)}$ but this doesn't seem helpful.
I'd appreciate any pointers or resources I can read to get better at this type of problem solving. Thank you!
Hint: $$\frac{k}{(k+1)!}=\frac{k+1-1}{(k+1)!}=\frac{1}{k!}-\frac{1}{(k+1)!}.$$ Now try summing a few terms.