(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)*(1+1/4)*.*(1+1/2004)*(1-1/2004)

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(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)*(1+1/4)*.*(1+1/2004)*(1-1/2004)

(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)*(1+1/4)*.*(1+1/2004)*(1-1/2004)
(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)*(1+1/4)*.*(1+1/2004)*(1-1/2004)

(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)*(1+1/4)*.*(1+1/2004)*(1-1/2004)
(1-1/2)*(1+1/2)*(1-1/3)*(1+1/3)*(1-1/4)*(1+1/4)*.*(1+1/2004)*(1-1/2004)
=(1/2)*(3/2)*(2/3)*(4/3)*.(2003/2004)*(2005/2004)
=(1/2)*(2005/2004)
=2005/4008