根据式子1/n(n+1)=(n+1)-n/n(n+1)=1/n-1/n+1,计算1/1x2+1/2x3+1/3x4+.+1/2007x2008.要有过程分析

题目

根据式子1/n(n+1)=(n+1)-n/n(n+1)=1/n-1/n+1,计算1/1x2+1/2x3+1/3x4+.+1/2007x2008.要有过程分析

答案

按提示,有
1/2 =1-1/2,
1/6=1/2-1/3,
1/12=1/3-1/4,
1/20=1/4-1/5,
.
1/(2007×2008)=1/2007-1/2008,
所以,1/(1x2)+1/(2x3)+1/(3x4)+.+1/(2007x2008)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+.+(1/2006-1/2007)+(1/2007-1/2008)
=1-1/2008
=2007/2008.