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