1/m(m+1)+1/(m+1)(m+2)+1/(m+2)(m+3)+.+1/(m+9)(m+10)

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1/m(m+1)+1/(m+1)(m+2)+1/(m+2)(m+3)+.+1/(m+9)(m+10)
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1/m(m+1)+1/(m+1)(m+2)+1/(m+2)(m+3)+.+1/(m+9)(m+10)
1/m(m+1)+1/(m+1)(m+2)+1/(m+2)(m+3)+.+1/(m+9)(m+10)

1/m(m+1)+1/(m+1)(m+2)+1/(m+2)(m+3)+.+1/(m+9)(m+10)
∵ 1/m(m+1) = 1/m - 1/(m+1),
1/(m+1)(m+2) = 1/(m+1) - 1/(m+2)
1/(m+2)(m+3) = 1/(m+2) - 1/(m+3)
所以原式 = 1/m - 1/(m+1) + 1/(m+1) - 1/(m+2) + 1/(m+2) - 1/(m+3) ...+ 1/(m+9) - 1/(m+10)
= 1 - 1/(m+10)
= (m+9)/(m+10)

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