find the sum
∑(n/(n+1)!) from n=1 to infinity
look at the pattern
sum(1) = 1/2 = (2! - 1)/2!
sum(2) = 1/2 + 2/6 = 5/6 = (3! - 1)/3!
sum(3) = 5/6 + 3/24 = 23/24 = (4! - 1)/4!
sum(4) = 23/24 + 4/120 = 119/120 = ....
...
sum(n) = ((n+1)! - 1)/(n+1)!