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