化简1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+2010)(a+2011)

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化简1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+2010)(a+2011)
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化简1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+2010)(a+2011)
化简1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+2010)(a+2011)

化简1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+2010)(a+2011)
1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+2010)(a+2011)
= 1/(a+1)-1/(a+2) + 1/(a+2)-1/(a+3) + .+1/(a+2010)-1/(a+2011)
= 1/(a+1)-1/(a+2011)
= 2010/[(a+1)(a+2011)]

这个有点技巧
1/(a+1)(a+2）等于1/(a+1）—1/(a+2），。。以此类推，最后是1/(a+1）—1/(a+2011)

1/(a+n)(a+n+1)=1/(a+n) - 1/(a+n+1)
所以原式=1/(a+1) - 1/(a+2) + 1/(a+2) - 1/(a+3) + ........+1/(a+2010)-1/(a+2011)
= 1/(a+1) - 1/(a+2011)

2010/a*a+2012a+2011

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