1/（2*4）+1/（4*6）+1/（6*8）+.+1/（2006*2008）
=(1/4)(1/(1*2)+1/(2*3)+.+1/(1003*1004))
=(1/4)(1-1/2+1/2-1/3+.+1/1003-1/1004)
=(1/4)(1-1/1004)
=1003/4016