Simplify:

Question

2/1! - 3/2! + 4/3! - 5/4! + ..... + 2010/2009! - 2011/2010!

Where k! is the multiplication of all numbers up to k (e.g. 5! = 5x4x3x2x1=120)

Answer 1

look for a pattern in the sum, let S(n) be the sum of the first n terms

S(1) = 2
S(2) = 2 - 3/2! =1/2
S(3) = 1/2 + 4/3! = 7/6
S(4) = 7/6 - 5/4! = 23/24
S(5) = 23/24 + 6/5! = 121/120

ahhh, ..

121/120 = (5! + 1)/5! = S(5)
23/24 = (4! - 1)/4! = S(4)
7/6 = (3! + 1)/3! = S(3)
1/2 = (2! - 1)/2! = S(2)
2 = (1! + 1)/1! = S(1)

so it looks like S(2010) = (2010! - 1)/2010!