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