How to simplify $\sum_{k=1}^nk\cdot k!$

Question

How do I go about simplifying this:

$$\sum_{k=1}^nk \cdot k!$$

Wolfram alpha tells me it's the same as $(n+1)!-1$ but I don't see how.

If true it must follow by induction. – David C. Ullrich Dec 01 '17 at 16:18 — David C. Ullrich, Dec 01 '17 at 16:18

score 14 · Answer 1 · answered Dec 01 '17 at 16:03

14

Hint: $$k\cdot k! = [(k+1)-1]\cdot k! = (k+1)!-k!$$

answered Dec 01 '17 at 16:03

nonuser

1 Answers1