How would I get the value of this sum in terms of n

Question

How would i evaluate this sum as an nth term:

$\sum_{r=1}^{n} r!r$

Also is there any particular method for simplifying sums that have an nth term with a factorial in?

I don't think there's a closed form for this sum. – 5xum Dec 06 '16 at 15:00 — 5xum, Dec 06 '16 at 15:00

score 6 · Answer 1 · answered Dec 06 '16 at 15:00

6

Hint. One may notice that $$ (r+1)!-r!=r\cdot r! $$

answered Dec 06 '16 at 15:00

Olivier Oloa

1

This is awesome. – Matias Heikkilä Dec 06 '16 at 15:02
so all of the factorials cancel out leaving just $(n+1)! - 1!$ which simplifies to $(n+1)! - 1$ – user6817585 Dec 06 '16 at 15:05
@user6817585 Yes, it's a telescoping sum. – Olivier Oloa Dec 06 '16 at 15:36

1 Answers1