1\1×3+1\3×5+1\5×7+1\7×9+1\11×13+1\13×15+1\15×17+.1\2007×2009
=1/2*(1/1-1/3+1/3-1/5+1/5-1/7+……+1/2007-1/2009)
=1/2*(1-1/2009)
=1/2*2008/2009
=1004/2009