+647 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.
Here is the closed form, but now you have to prove it !!!
S(n) =[n + 1]! - 1
Hint: nn! = [n + 1]! - n!
+129852
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
