H = 1.1!2.2! + ... +n.n!

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

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

= 1.2.3.....n.(n + 1) - n!

= (n + 1)! - n!

H = 1.1! + 2.2! + 3.3! + ... + n.n!

= 2! - 1! + 3! - 2! + 4! - 3! + ... + (n + 1)! - n!

= - 1! + (n + 1)!

= (n + 1)! - 1

Ta có:

2!-1!= 1!.( 2-1)= 1!

3!-2!= 2!.( 3-1)= 2.2!

4!-3!= 3!.( 4-1)= 3.3!

....

⇒ ( n+1)!-n!= n!.( n+1-1)= n.n!

Do đó tổng H= 1!+2.2!+3.3!+....+n.n!

= 2!-1!+3!-2!+4!-3!+...+( n+1)!-n!

= ( n+1)!-1!

H=1.1!+2.2!.......................n.n!

=(2−1)1!+(3−1)2!....................(n+1−1)n!

=2.1!−1.1!+3.2!−2!................(n+1)n!−n!

=2!−1!+3!−2!.........................(n+1)n!−n!

=(n+1)!−1