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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