计算:1/x(x+1) + 1/(x+1)(x+2) + 1/(x+2)(x+3) +.+1/(x+2010)(x+2011)
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计算:1/x(x+1) + 1/(x+1)(x+2) + 1/(x+2)(x+3) +.+1/(x+2010)(x+2011)
1/x(x+1) + 1/(x+1)(x+2) + 1/(x+2)(x+3) +.+1/(x+2010)(x+2011)
=1/x-1/(x+1) + 1/(x+1)-1/(x+2) + 1/(x+2)-1/(x+3) +.+1/(x+2010)-1/(x+2011)
=1/x-1/(x+2011)
=2011/x(x+2011)
1/x(x+1) + 1/(x+1)(x+2) + 1/(x+2)(x+3) +.......+1/(x+2010)(x+2011)
=(1/x-1/(x+1) )+ (1/(x+1)-1/(x+2)+ (1/(x+2)-1/(x+3) )+……+(1/(x+2010)-1/(x+2011))
=1/x-1/(x+2011)
=(x+2011-x)/(x*(x+2011) )
=2011/(x(x+2011))
就是1/x(x+1)=1/x-1/x+1
1/(x+1)(x+2) =1/(x+1)-1/(x+2)
.......
原式=1/x-1/(x+1)+1/(x+1)-1/(x+2) +。。。+1/(x+2010)-1/(x+2011)
=1/x-1/(x+2011)
=2011/x(x+2011)
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