calculate sum k!*k for k=1 to k=n

Sum k!*k = 1!*1 + 2!*2 + ... +
n!*n

We can add and subtract 1, so
that:

Sum k!*(k + 1 - 1) = Sum k!*(k+1) - Sum
k!

But k!*(k+1) = (k+1)!

We'll
re-write the sum:

Sum k!*(k+1) - Sum k! = Sum (k+1)! - Sum
k!

Sum (k+1)! = 2! + 3! + ... + n! + (n+1)!
(1)

Sum k! = 1! + 2! + 3! + ... + n!
(2)

We'll subtract (2) from
(1):

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

We'll eliminate like
terms:

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

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