数学中国

标题: 求极限 lim(n→∞)an ，其中 an={ ∏(k=1,n)(kn)^[1／(n+k)] }^[1／ln(n)] [打印本页]

作者: elim 时间: 2023-12-31 11:31
标题: 求极限 lim(n→∞)an ，其中 an={ ∏(k=1,n)(kn)^[1／(n+k)] }^[1／ln(n)]
题: 计算 \(\displaystyle\lim_{n\to\infty}\bigg(\prod_{k=1}^{n}(kn)^{\frac{1}{n+k}}\bigg)^{\frac{1}{\ln n}}\)

作者: elim 时间: 2024-1-1 10:16
本帖最后由 elim 于 2024-1-1 02:06 编辑

题：求\(\displaystyle\lim_{n\to\infty}a_n,\text{ 其中}\,a_n=\bigg(\prod_{k=1}^{n}(kn)^{\frac{1}{n+k}}\bigg)_{\dot\,}^{\frac{1}{\ln n}}\)
解： \(\because\;\ln a_n=\displaystyle\sum_{k=1}^n\frac{\ln\frac{k}{n}+2\ln n}{(k+n)\ln n}={\small\frac{1}{\ln n}\frac{1}{n}}\sum_{k=1}^n\frac{\ln\frac{k}{n}}{1+\frac{k}{n}}+2\sum_{k=n+1}^{2n}\frac{1}{k}\)
\(\qquad\quad\small\displaystyle\frac{1}{n}\sum_{k=1}^n\frac{|\ln\frac{k}{n}|}{1+\frac{k}{n}}\to \int_0^1\frac{|\ln x|}{1+x}dx < \int_0^1|\ln x| dx =1,\;\sum_{k=n+1}^{2n}\frac{2}{k}\to 2\ln 2\)
\(\qquad\therefore\displaystyle\lim_{n\to\infty}a_n = \exp(2\ln 2)=4.\quad\square\)

注记： \(\lim_n(\sum_1^{2n}k^{-1}-\sum_1^n k^{-1})=\lim_n(\ln 2n-\ln n)=\ln 2;\)
\(\qquad\quad\displaystyle\int_0^1\frac{\ln x}{1+x}dx=-\frac{\pi^2}{12};\)
\(\qquad\quad\ln a_n -2\ln 2=O(1/\ln n).\)

作者: elim 时间: 2024-1-6 04:14
@王守恩，Mathematica 能不能算这个极限？

作者: 王守恩 时间: 2024-1-6 07:38

Table[N[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[\(n*k\), \(n + k\)]\))^(1/Log[n])], {n, 9, 19}]

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{2.88141, 2.91218, 2.93899, 2.96263, 2.98370, 3.00264, 3.01980, 3.03544, 3.04979,3.06302,3.07527}

Table[N[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[\(n*k\), \(n + k\)]\))^(1/Log[n])], {n, 99, 109}]

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{3.35123, 3.35246, 3.35367, 3.35487, 3.35605, 3.35721, 3.35836, 3.35949, 3.36061,3.36172,3.36281}

Table[N[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[\(n*k\), \(n + k\)]\))^(1/Log[n])], {n, 999, 1009}]

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{3.55141, 3.55148, 3.55154, 3.55160, 3.55166, 3.55172, 3.55178, 3.55184, 3.55190,3.55196,3.55202}

Table[N[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[\(n*k\), \(n + k\)]\))^(1/Log[n])], {n, 9999, 10009}]

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{3.65832, 3.65833, 3.65833, 3.65833, 3.65834, 3.65834, 3.65835, 3.65835,3.65835,3.65836,3.65836}

Table[N[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[\(n*k\), \(n + k\)]\))^(1/Log[n])], {n, 99999, 100009}]

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{3.72422, 3.72422, 3.72422, 3.72422, 3.72422, 3.72422, 3.72422, 3.72422,3.72422,3.72422,3.72422}

Table[N[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[\(n*k\), \(n + k\)]\))^(1/Log[n])], {n, 999999, 999999}]

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{3.76882}

Limit[(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*RadicalBox[(n*k\), \(n + k\)]\))^(1/Log[n]), n -> \[Infinity]]

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\!\(\*UnderscriptBox[\(\[Limit]\), \(n \[Rule] \[Infinity]\)]\)
(\!\(\*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(n\)]
\*SuperscriptBox[\((k\ n)\), FractionBox[\(1\), \(k + n\)]]\))^(1/Log[n])
\(\displaystyle\lim_{n\to\infty}\bigg(\prod_{k=1}^{n}\sqrt[k+n]{kn}\bigg)^{\frac{1}{\ln(n)}}=?\) 不能算这个极限?

作者: elim 时间: 2024-1-6 14:12

王守恩发表于 2024-1-5 16:38
{2.88141, 2.91218, 2.93899, 2.96263, 2.98370, 3.00264, 3.01980, 3.03544, 3.04979,3.06302,3.07527}
...

这东西收敛太慢，不是mathematica 的莱．

作者: luyuanhong 时间: 2024-1-7 08:46
楼上 elim 的帖子很好！已收藏。

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