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计算1/1*3+1/3*5+1/5*7+...+1/(2n-3)(2n-1)+1/(2n-1)(2n+1)

分类: 作业答案 • 2021-12-19 10:51:52
问题描述：

计算1/1*3+1/3*5+1/5*7+...+1/(2n-3)(2n-1)+1/(2n-1)(2n+1)

答

1/1*3+1/3*5+1/5*7+...+1/(2n-3)(2n-1)+1/(2n-1)(2n+1)
=1/2*(1-1/3)+1/2*(1/3-1/5)+1/2*(1/5-1/7)+...+1/2*(1/(2n-3)-1/(2n-1)) +1/2*(1/(2n-1)-1/(2n+1))
=1/2*(1-1/(2n+1))
=n/(2n+1)

答

=1/2(1-1/3+1/3-1/5+1/5-1/7+……1/(2n-3)-1/(2n-1)+1/(2n-1)-1/(2n+1))
=1/2(1-1/(2n+1))
=1/2-1/(4n+2)

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