Question

1/1x2+1/2x3+1/3x4+.1/2007x2008

Answer 1

1/(n×(n+1))=(n+1-n)/(n×(n+1))=(n+1)/(n×(n+1))-n/(n×(n+1))=1/n-1/(n+1)

所以

1/1x2=1/1-1/2

1/2x3= 1/2-1/3

1/3x4= 1/3-1/4

……

1/2006×2007=1/2006-1/2007

1/2007×2008= 1/2007-1/2008

全部加起来

只剩下 1-1/2008=2007/2008

最终结果为2007/2008