（-1x1/2）+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)

（-1x1/2）+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)
（-1x1/2）+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)

（-1x1/2）+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)
（-1x1/2）+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009）
=-｛（1x1/2）+(1/2x1/3)+(1/3x1/4)+.+(1/2007x1/2008)+(1/2008X1/2009)｝
=-｛1-1/2+1/2-1/3+1/3-1/4+.+1/2007-1/2008+1/2008-1/2009}
=-(1-1/2009)
=-2008/2009
补充：1/n*1/(n+1)=1/n-1/(n+1) 例如1/2*1/3=1/2-1/3=1/6

（-1x1/2）+(-1/2x1/3)+(-1/3x1/4)+....+(-1/2007x1/2008)+(-1/2008X1/2009)
= -1×（1-1/2+1/2-1/.3+……+1/2008-1/2009）
= -1×2008/2009
= -2008/2009