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1X1/2+1/2X1/3+1/3X1/4+1/4X1/5+1/5X1/6+1/6X1/7,...

原式=1/2+1/6+1/12+1/20+1/30+1/42 =(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7) =1-1/7 =6/7 公式: 1/a乘1/(a+1)=1/a-1/(a+1)其中a是正整数

1x1/2+1/2x1/3+1/3x1/4+1/4x1/5+1/5x1/6+1/6x1/7 =1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/9-1/7 =1-1/7 =6/7

提公因式、通分,就容易计算了。具体见下图。

利用裂项相消法:1/nx1/(n+1)=1/n-1/(n+1) (-1x1/2)+(-1/2x1/3)+(-1/3x1/4)+…+(-1/2015x1/2016) =-[(1x1/2)+(1/2x1/3)+(1/3x1/4)+…+(1/2015x1/2016)] =-(1-1/2+1/2-1/3+1/3-1/4+,,,,,,,+1/2015-1/2016) =-(1-1/2016) =-2015/2016

请采纳

我们可以把这个式看成 (1-1/2)+(1-1/3)+……+(1-1/100) =1×100-(1/2+1/3+……1/100) =100-(5049/200)【首项加末项乘以项数除以2】 =14951/200

如图:首先仔细观察,寻找规律,单项分解得到 ((n-1)(n+1)-3)/(n-1)(n+1)=((n^2)-1-3)/(n-1)(n+1)=(n-2)(n+2)/(n-1)(n+1) 代入原式,不难发现前后各项分子分母递增,可互相消去。 原式=101/(4X98)=101/392

=1-1/2+1/2-1/3+1/3-1/4+...+1/2014-1/2015 =1-1/2015 =2014/2015

=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7 =1-1/7 =6/7

你第三个是不是3*1/4啊? 如果是我们可以把这个式看成 (1-1/2)+(1-1/3)+……+(1-1/100) =1×100-(1/2+1/3+……1/100) =100-(5049/200)【首项加末项乘以项数除以2】 =14951/200

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