发现的规律是
-1/[n*(n+1)]= -1/n +1/(n+1)
所以(-1X2分之1)= -1 +1/2
(-2分之1X3分之1)= -1/2 +1/3
以此类推
(-2007分之1X2008分之1)= -1/2007 +1/2008
于是
(-1X2分之1)+(-2分之1X3分之1)+(-3分之1X4分之1)+……+(-2007分之1X2008分之1)
= -1 +1/2 -1/2 +1/3 -1/3 +1/4 +……-1/2007 + 1/2008
= -1 +(1/2 -1/2) +(1/3 -1/3) + …… +(1/2007 -1/2007) + 1/2008
= -1 +1/2008
= -2007/2008