Find the sum of 1x1!+2x2!+3x3!+4x4!+5x5!+..........+nxn!

actaully (n+1)!-1 is real ans

how to proff?

Let’s try when n=5,
1=2-1
2=3-1 etc.

(2-1)1! + (3-1)2! + (4-1)3! + (5-1)4! + (6-1)5!
= (2x1! – 1x1!) + (3x2! – 1x2!) + (4x3! – 1x3!) + (5x4! – 1x4!) + (6x5!-1x5!)

Notice that:
2x1! = 2!, 3x2! = 3! etc.

Thus,
=2! – 1! + 3! – 2! + 4! – 3! + 5! – 4! + 6! - 5!

 

All of them are eliminated, left with 6! -1!

Thus when n = 5, this series = (5+1)!-1

(n+1)!-1.