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