1/2*3/4*5/6*.*2n-1/2n
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1/2*3/4*5/6*.*2n-1/2n1/2*3/4*5/6*.*2n-1/2n1/2*3/4*5/6*.*2n-1/2n当n=1时,左=1/2∵[1/2*3/4*5/6*…(2n-1)/2n]^
1/2*3/4*5/6*.*2n-1/2n
1/2*3/4*5/6*.*2n-1/2n
1/2*3/4*5/6*.*2n-1/2n
当n=1时,左=1/2
∵[1/2*3/4*5/6*…(2n-1)/2n]^2
=[1/2*3/4*5/6*…(2n-1)/2n]×[1/2*3/4*5/6*…(2n-1)/2n]
又∵(2n)^2>(2n)^2-1,即(2n)^2>(2n-1)(2n+1)
∴(2n-1)/2n<2n/(2n+1)
故:1/2<2/3 3/4<4/5......以此类推(2n-1)/2n<2n/(2n+...
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∵[1/2*3/4*5/6*…(2n-1)/2n]^2
=[1/2*3/4*5/6*…(2n-1)/2n]×[1/2*3/4*5/6*…(2n-1)/2n]
又∵(2n)^2>(2n)^2-1,即(2n)^2>(2n-1)(2n+1)
∴(2n-1)/2n<2n/(2n+1)
故:1/2<2/3 3/4<4/5......以此类推(2n-1)/2n<2n/(2n+1)
原式<[1/2*3/4*5/6*…(2n-1)/2n]×[2/3*4/5*6/7*…2n/(2n+1)]
=1/(2n+1)中间全部约分了。
∵不等式两边均为正数,同时开方,可得:
1/2*3/4*5/6*…2n-1/2n<1/根号(2n+1)
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