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