1/1*2+1/2*3+1/3*4+...+1/2006*2007=?

1/1*2+1/2*3+1/3*4+...+1/n(n+1)
=1-1/2+1/2-1/3+1/3-1/4+...+1/n-1/(n+1)
=1-1/(n+1)
=n/(n+1)
1/1*2+1/2*3+1/3*4+...+1/2006*2007
根据上面那个公式把n=2006代入就可以得到
=2006/2007
1/2*4+1/4*6+1/6*8+...+1/2006*2008
=1/2(1/2-1/4+1/4-1/6+1/6-1/8+...+1/2006-1/2008)
=1/2(1/2-1/2008)
=1/2*1003/2008
=1003/4016

2006/2007
n/(n+1)
1003/4016

易证1/n(n+1)=1/n-1/(n+1)
（1）原式=1/1-1/2+1/2-1/3+……+1/2006-1/2007=1-1/2007=2006/2007
(2)原式=1/1-1/2+1/2-1/3+……+1/n-1/(n+1)=1-1/(n+1)=n/(n+1)
(3)原式=0.25(1/1*2+1/2*3+1/3*4+……1/1003*1004)=0.25(1-1/1004)=1003/4016

.......sorry ~

1-1/2+1/2-1/3...+1/2006-1/2007=2006/2007
同理，这叫裂项。1-1/(n+1)
1/2-1/2008=1003/2008