(-1x1/2)+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)怎么算?

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(-1x1/2)+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)怎么算?
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(-1x1/2)+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)怎么算?
(-1x1/2)+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)怎么算?

(-1x1/2)+(-1/2x1/3)+(-1/3x1/4)+.+(-1/2007x1/2008)+(-1/2008X1/2009)怎么算?
首先把负号提出来,则有
1x1/2=1-1/2
1/2x1/3=1/2-1/3
1/3x1/4=1/3-1/4
.
1/2008X1/2009=1/2008-1/2009
将以上各式相加,得原式=-（1-1/2009）=-2008/2009

原式=(1/2-1)+(1/3-1/2)+(1/4-1/3)+...,,,+(1/2008-1/2007)+(1/2009-1/2008)
括号内各式前后消除
原式=1/2009-1