2項係数の相加平均・相乗平均を含む極限

2項係数の相加平均・相乗平均を含む極限

(1)

\[ \lim_{n\rightarrow\infty}\sqrt[n]{\frac{1}{n+1}\sum_{k=0}^{n}C\left(n,k\right)}=2 \]

(2)

\[ \lim_{n\rightarrow\infty}\sqrt[n]{\sqrt[n+1]{\prod_{k=0}^{n}C\left(n,k\right)}}=\sqrt{e} \]

-

\(C\left(x,y\right)\)は2項係数

(1)

\begin{align*} \lim_{n\rightarrow\infty}\sqrt[n]{\frac{1}{n+1}\sum_{k=0}^{n}C\left(n,k\right)} & =\lim_{n\rightarrow\infty}\sqrt[n]{\frac{1}{n+1}\sum_{k=0}^{n}C\left(n,k\right)1^{k}1^{n-k}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n]{\frac{2^{n}}{n+1}}\\ & =2\lim_{n\rightarrow\infty}\left(1+n\right)^{-\frac{1}{n}}\\ & =2\lim_{n\rightarrow\infty}e^{-\frac{\log\left(1+n\right)}{n}}\\ & =2 \end{align*}

(2)

\begin{align*} \lim_{n\rightarrow\infty}\sqrt[n]{\sqrt[n+1]{\prod_{k=0}^{n}C\left(n,k\right)}} & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\prod_{k=0}^{n}\frac{n!}{k!\left(n-k\right)!}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\left(n!\right)^{n+1}\prod_{k=0}^{n}\left(\frac{1}{k!}\right)^{2}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\left(n!\right)^{n+1}\prod_{k=1}^{n}\left(\frac{1}{k!}\right)^{2}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\left(n!\right)^{n+1}\prod_{k=1}^{n}\prod_{j=1}^{k}\left(\frac{1}{j}\right)^{2}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\left(n!\right)^{n+1}\prod_{j=1}^{n}\prod_{k=1}^{n+1-j}\left(\frac{1}{j}\right)^{2}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\prod_{j=1}^{n}j^{n+1}\prod_{j=1}^{n}\left(\frac{1}{j^{n+1-j}}\right)^{2}}\\ & =\lim_{n\rightarrow\infty}\sqrt[n\left(n+1\right)]{\prod_{j=1}^{n}j^{2j-n-1}}\\ & =\lim_{n\rightarrow\infty}\exp\left(\frac{1}{n\left(n+1\right)}\sum_{j=1}^{n}\left(\left(2j-n-1\right)\log j\right)\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(\frac{1}{n\left(n+1\right)}\sum_{j=1}^{n}\left(\left(2j-n\right)\left(\log j-\log n\right)+\left(2j-n\right)\log n-\log j\right)\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(\frac{1}{n\left(n+1\right)}\left(\sum_{j=1}^{n}\left(2j-n\right)\left(\log j-\log n\right)+n\log n-\sum_{j=1}^{n}\log j\right)\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(\frac{1}{n\left(n+1\right)}\left(\sum_{j=1}^{n}\left(2j-n\right)\left(\log j-\log n\right)\right)\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(\frac{1}{\left(n+1\right)}\sum_{j=1}^{n}\left(\left(2\frac{j}{n}-1\right)\log\frac{j}{n}\right)\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(\frac{n}{\left(n+1\right)}\frac{1}{n}\sum_{j=1}^{n}\left(\left(2\frac{j}{n}-1\right)\log\frac{j}{n}\right)\right)\\ & =\exp\left(\int_{0}^{1}\left(2x-1\right)\log xdx\right)\\ & =\exp\left(\left[\left(x^{2}-x\right)\log x\right]_{0}^{1}-\int_{0}^{1}\left(x-1\right)dx\right)\\ & =\exp\left(-\left[\frac{1}{2}x^{2}-x\right]_{0}^{1}\right)\\ & =\sqrt{e} \end{align*}

ページ情報

タイトル

2項係数の相加平均・相乗平均を含む極限

URL

https://www.nomuramath.com/qbrevixr/

SNSボタン