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

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1/1x2+1/2x3+1/3x4+.1/2007x20081/1x2+1/2x3+1/3x4+.1/2007x20081/1x2+1/2x3+1/3x4+.1/2007x20081/(n×(n+1)

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

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