let
x =1/2+1/3+...+1/2009
(1-1/2-1/3-...-1/2009)(1/2+1/3+1/4...+1/2010)-(1-1/2-1/3-...-1/2009-1/2010)(1/2+1/3...+1/2009)
= (1-x)(x+1/2010)- (1-x-1/2010)x
= [x+1/2010 - x^2 - x/2010] - [ x- x^2 -x/2010]
= 1/2010