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