I'm having some issues with the inductive step for this problem, and I could use a hand with it.

Let $\displaystyle F(n)$ be the statement that $\displaystyle 1\cdot1!+2\cdot2!+...+n\cdot n!=(n+1)!-1$ whenever $\displaystyle n$ is a positive integer.

I have the basis step and inductive hypothesis done, but I'm having some trouble with the induction proof for $\displaystyle F(k+1)$. Does anyone have any advice?