Proving assertion with and without induction

Question

I have successfully proven $ \displaystyle \sum_{k=1}^n k ·(k!) = (n+1)! -1 $ with mathematical induction for all $n \in \mathbb{N}$. Now, how would someone prove this assertion without induction?

Answer 1

Hint : $$k\cdot k!=(k+1)\cdot k!-k!=(k+1)!-k!$$ Almost all terms will cancel out.