1/2+5/6+11/12+.+109/110

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1/2+5/6+11/12+.+109/110

1/2+5/6+11/12+.+109/110
1/2+5/6+11/12+.+109/110

1/2+5/6+11/12+.+109/110
利用1/[n*(n+1)]=1/n - 1/(n+1) 进行简便计算,计算过程如下:
原式=(1-1/2)+(1-1/6)+(1-1/12)+...+(1-1/110)
=10-(1/2+1/6+1/12+...1/110)
=10-[1/(1*2)+1/(2*3)+1/(3*4)+...+1/(10*11)]
=10-(1- 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ...+ 1/10 - 1/11)
=10-(1-1/11)
=10-10/11
=100/11

1/2+5/6+11/12+.......+109/110
=1-1/2+1-1/6+1-1/12+.......+1-1/110
=1-1/(1*2)+1-1/(2*3)+1-1/(3*4)+......+1-1/(10*11)
=1*10-(1-1/2+1/2-1/3+1/3-1/4+......+1/10-1/11)
=10-10/11
=100/11

1/2+5/6+11/12+19/20+……+89/90+109/110
=(1-1/2)+(1-1/6)+(1-1/12)+(1-1/20)+.......+(1-1/90)+(1-1/110)
=1-1/2+1-1/6+1-1/12+1-1/20+......+1-1/90+1-1/110
=1-(1-1/2)+1-(1/2-1/3)+1-(1/3-1/4)+1-...

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1/2+5/6+11/12+19/20+……+89/90+109/110
=(1-1/2)+(1-1/6)+(1-1/12)+(1-1/20)+.......+(1-1/90)+(1-1/110)
=1-1/2+1-1/6+1-1/12+1-1/20+......+1-1/90+1-1/110
=1-(1-1/2)+1-(1/2-1/3)+1-(1/3-1/4)+1-(1/4-1/5)+......+1-(1/9-1/10)+1-(1/10-1/11)
=1*9-1+1/2-1/2+1/3-1/3+1/4-1/4+1/5+......-1/9+1/10-1/10+1/11(因为从2开始到11结束,所以有9项)
=9-1+1/11
=8又1/11

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