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若(1/1x3)+(1/3x5)+(1/5x7)+……+[1/(2n-1)(2n+1]=?用含有n的式子表示
题目详情
若(1/1x3)+(1/3x5)+(1/5x7)+……+[1/(2n-1)(2n+1]=?用含有n的式子表示
▼优质解答
答案和解析
1/(2n-1)(2n+1)
=1/2*2/(2n-1)(2n+1)
=1/2*[(2n+1)-(2n-1)]/(2n-1)(2n+1)
=1/2*[(2n+1)/(2n-1)(2n+1)-(2n-1)/(2n-1)(2n+1)]
=1/2*[1/(2n-1)-1/(2n+1)]
所以原式=1/2*[1-1/3+1/3-1/5+……+1/(2n-1)-1/(2n+1)]
=1/2*[1-1/(2n+1)]
=n/(2n+1)
=1/2*2/(2n-1)(2n+1)
=1/2*[(2n+1)-(2n-1)]/(2n-1)(2n+1)
=1/2*[(2n+1)/(2n-1)(2n+1)-(2n-1)/(2n-1)(2n+1)]
=1/2*[1/(2n-1)-1/(2n+1)]
所以原式=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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