1/(1×3)+1/(3×5)+1/(5×7)+.+1/(2n-1)(2n+1)等于多少?

来源：学生作业帮助网编辑：六六作业网时间：2024/12/27 01:06:30

1/(1×3)+1/(3×5)+1/(5×7)+.+1/(2n-1)(2n+1)等于多少?1/(1×3)+1/(3×5)+1/(5×7)+.+1/(2n-1)(2n+1)等于多少?1/(1×3)+1/

1/(1×3)+1/(3×5)+1/(5×7)+.+1/(2n-1)(2n+1)等于多少?
1/(1×3)+1/(3×5)+1/(5×7)+.+1/(2n-1)(2n+1)等于多少?

1/(1×3)+1/(3×5)+1/(5×7)+.+1/(2n-1)(2n+1)等于多少?
=1/2（1-1/3+1/3-1/5+…… +1/（2n-1）-1/（2n+1））
1/2（1-1/（2n+1））=n/2n+1

1/(2n-1)(2n+1)=[1/(2n-1)-1/(2n+1)]/2
1/(1×3)+1/(3×5)+1/(5×7)+......+1/(2n-1)(2n+1)
=1/2[1-1/3+1/3-1/5+1/5-1/7+......+1/(2n-1)-1/(2n+1)]
=1/2[1-1/(2n+1)]
=n/(2n+1)

原式=(1/2)[(1-1/3)+(1/3-1/5)+(1/5-1/7)+...+(1/(2n-1)-1/(2n+1))]
=(1/2)[1-1/(2n+1)]
=n/(2n+1)

把an转换
an=1/(2n-1)(2n+1)=1/2[1/(2n-1)-1/(2n+1)]
则原式=1/2[1-1/3+1/3-1/5+…… +1/（2n-1）-1/（2n+1）]
=1/2[1-1/（2n+1）]
=n/(2n+1)

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