1/1*2+1/2*3+1/3*4+1/4*5+.........1/2007*2008

来源：百度知道 编辑：UC知道 时间：2024/05/31 02:17:20
要过程哦！

因为1/1*2 = 1/1 - 1/2,
1/2*3 = 1/2 - 1/3,
1/3*4 = 1/3 - 1/4,
...(省略)
1/2007*2008 = 1/2007 - 1/2008,

所以
1/1*2 + 1/2*3 + 1/3*4 + 1/4*5 + ... + 1/2007*2008
=(1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ... + (1/2007 - 1/2008)
=1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/2007 - 1/2008
=1/1 + (1/2 - 1/2) + (1/3 - 1/3) + (1/4 - 1/4) + (1/5 - 1/5) + ... + (1/2007 - 1/2007) - 1/2008
=1 - 1/2008
=2007/2008

1/1*2=1/2=1-1/2
1/2*3=1/6=1/2-1/3
1/3*4=1/12=1/3-1/4
.......以此类推

于是原式=(1-1/2)+(1/2-1/3)+(1/3-1/4)+.......+(1/2007-1/2008)
消掉相同的式子
=1-1/2008
=2007/2008

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