1/2*1/3+1/3*1/4+1/4*1/5+……+1/2006*1/2007+1/2007*1/2008
=1/2-1/3+1/3-1/4+1/4-1/5+……+1/2006-1/2007+1/2007-1/2008
=1/2-1/2008
=1003/2008