k! = 1*2*3*...*k

(k - 1)/k! = k/k! - 1/k!

=> 1/(k - 1)! - 1/k!

The sum of (k - 1)/k! for k = 1 to n is:

1/0! - 1/1! + 1/1! - 1/2! + 1/2! - 1/3! + ... + 1/(n - 1)! - 1/n!

=> 1/0! - 1/n!

substitute 0! = 1

=> 1/1 - 1/n!

=> (n! - 1)/n!

**The required sum is (n! - 1)/n!**

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