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丨1-1/2丨+丨1/2-1/3/丨+|1/3-1/4|+……+|1/2013-1/2...

=1-1╱2014=2013╱2014

中间各项相互抵消

-1/4+1/2ⅹ(2/3十丨2/3-2丨) =-1/4+1/2*(2/3+4/3) =-1/4+1/2*2 =-1/4+1 =3/4

原式=1-1/2015 =2014/2015

简便计算方法: 1+2+3+...+n=n(n+1)/21/(1+2+3+...+n)=2/n(n+1)=2[1/n-1/(n+1)]1+1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+......+1/(1+2+3+...+100)=2[(1-1/2)+(1/2-1/3)+(1/3-1/4)+......+(1/100-1/101)]=2(1-1/101)=200/101 它的原理是根据公式:1/n(n...

原式 =-2/3-1/3-1/12-3 =-1-3-1/12 =-4又1/12

=1-1/2+1/2-1/3+1/3-1/4+....+1/2006-1/2007 =1-1/2007 =2006/2007

丨3/1-2/1丨+丨4/1-3/1丨+丨5/1-4/1丨+···+丨10/1-9/1丨 =1/2-1/3+1/3-1/4+1/4-1/5+......+1/9-1/10 =1/2-1/10 =2/5

1/(1十2)十1/(1+2+3)+....1/(1+2+3+4+....+99) =2/(2×3)+2/(3×4)+……+2/(99×100) =2×(1/2-1/3+1/3-1/4+……+1/99-1/100) =2×(1/2-1/100) =1-1/50 =49/50

(1+1/2)*(1+1/3)*(1+1/4)*.....*(1+1/19)*(1+1/20) = 3/2*4/3*5/4*......*20/19*21/20 = 21/2 【备注:前一个分数的分子与后一个分数的分母依次相约,最后只剩下第一个分数的分母和最后一个分数的分子】

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