当a=-2,b=-1时,1/ab+1/(a-1)*(b-1)+1/(a-1)(b-1)+.+1/(a-2012)(b-2012)分母都是一

来源:学生作业帮助网 编辑:作业帮 时间:2024/04/29 23:04:55
当a=-2,b=-1时,1/ab+1/(a-1)*(b-1)+1/(a-1)(b-1)+.+1/(a-2012)(b-2012)分母都是一

当a=-2,b=-1时,1/ab+1/(a-1)*(b-1)+1/(a-1)(b-1)+.+1/(a-2012)(b-2012)分母都是一
当a=-2,b=-1时,1/ab+1/(a-1)*(b-1)+1/(a-1)(b-1)+.+1/(a-2012)(b-2012)
分母都是一

当a=-2,b=-1时,1/ab+1/(a-1)*(b-1)+1/(a-1)(b-1)+.+1/(a-2012)(b-2012)分母都是一
A=-2 ,B=-1
1/AB+1/(A-1)(B-1)+1/(A-2)(B-2)+……+1/(A-2012)(B-2012)
=1/(-2)*(-1)+1/(-2-1)*(-1-1)+1/(-2-2)*(-1-2)……+1/(-2-2010)*(-1-2010)
=1/2+1/6+1/12+……+1/2014*2013
=1-1/2+1/2-1/3+1/3-1/4+……+1/2011-1/2012
=1-1/2014
=2013/2014

本题用的是拆分法计算的:公式为
(A+B)/AB=1/A+1/B

1/ab+1/(a-1)*(b-1)+1/(a-1)(b-1)+.......+1/(a-2012)(b-2012) 上式带入a=-2,b=-1得1/(2*1)+1/(3*2)+...+1/(2014*2013)=(1-1/2)+(1/2-1/3)+....+(1/2013-1/2014)=1-1/2014=2013/2014

裂项相加,有一部分可以对消。

题目可以变为1/(1*2)+1/(2*3)+......+1/(2013*2014)
=1/1-1/2+1/2-1/3+1/3-1/4.........+1/2013-1/2014
=2013/2014

1/ab+1/(a-1)*(b-1)+1/(a-2)(b-2)+.......+1/(a-2012)(b-2012)
=1/(-2)×(-1)+1/(-3)×(-2)+1/(-4)×(-3)+……+/(-2014)×(-2013) 代入
=1/1×2+1/2×3+1/3×4+……+1/2013×2014
=1-1/2+1/2-1/3+1/3-1/4+……+1/2013-1/2014
=1-1/2014
=2013/2014
数列的小错位相减

1/ab+1/(a-1)*(b-1)+1/(a-2)(b-2)+.......+1/(a-2012)(b-2012)
=1/(-2)×(-1)+1/(-3)×(-2)+1/(-4)×(-3)+……+/(-2014)×(-2013) 代入
=1/1×2+1/2×3+1/3×4+……+1/2013×2014
=1-1/2+1/2-1/3+1/3-1/4+……+1/2013-1/2014
=1-1/2014
=2013/2014