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证明1/(123)+1/(234)+1/(345）.....+1/n（n+1)(n+2)<1/4

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2 | 2011-7-3 19:49:03 | 显示全部楼层 |阅读模式

1/(1*2*3)=(1/2)*[1/(1*2)-1/(2*3)]1/(2*3*4)=(1/2)*[1/(2*3)-1/(3*4)]...1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+...+1/n(n+1)(n+2)=(1/2)*[1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+1/(3*4)-1/(4*5)+...+1/n(n+1)-1/(n+1)(n+2)]=(1/2)*[1/2-1/(n+1)(n+2)]=1/4-1/2(n+1)(n+2)所以原式一定小于1/4...

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