Find a closed form for

\(S_n = 1 \cdot 1! + 2 \cdot 2! + \ldots + n \cdot n!.\)

for integer n >= 1 Your response should have a factorial.

waffles
Feb 14, 2018

#1**+1 **

Here is the closed form, but now you have to prove it !!!

**S(n) =[n + 1]! - 1**

Hint: nn! = [n + 1]! - n!

Guest Feb 14, 2018

#2**+1 **

I'm stealing something here that I learned from heureka.....so, really....he should get the credit !!!!

He noted that

n * n! =

( [ n + 1 ] - 1) * n! =

(n + 1)n! - n! =

(n + 1)! - n!

So we have the following

1 * 1! = 2! - 1!

+ 2 * 2! = 3! - 2!

+ 3 * 3! = 4! - 3!

+ 4 * 4! = 5! - 4!

+ .......

+ n * n! = (n + 1)! - (n)!

The terms in red will "cancel" and we will be left with

Sum = ( n + 1)! - 1! =

Sum = (n + 1)! - 1

CPhill
Feb 14, 2018