中央2項係数の総和
中央2項係数の総和
\[ \sum_{k=0}^{\infty}C^{-1}\left(2k,k\right)=\frac{4}{3}+\frac{2\sqrt{3}\pi}{27} \]
\[ \sum_{k=0}^{\infty}C^{-1}\left(2k,k\right)=\frac{4}{3}+\frac{2\sqrt{3}\pi}{27} \]
\begin{align*}
\sum_{k=0}^{\infty}C^{-1}\left(2k,k\right) & =\sum_{k=0}^{\infty}\frac{(k!)^{2}}{(2k)!}\\
& =\sum_{k=0}^{\infty}\frac{\Gamma(k+1)\Gamma(k+1)}{\Gamma(2k+1)}\\
& =1+\sum_{k=1}^{\infty}\frac{\Gamma(k+1)\Gamma(k+1)}{\Gamma(2k+1)}\\
& =1+\sum_{k=1}^{\infty}\frac{k\Gamma(k)\Gamma(k+1)}{\Gamma(2k+1)}\\
& =1+\sum_{k=1}^{\infty}kB(k,k+1)\\
& =1+\sum_{k=1}^{\infty}k\int_{0}^{1}t^{k-1}(1-t)^{k}dt\\
& =1+\sum_{k=1}^{\infty}k\int_{0}^{1}\frac{1}{t}(t-t^{2})^{k}dt\\
& =1+\sum_{k=1}^{\infty}\int_{0}^{1}\frac{1}{t}\left(t-t^{2}\right)\frac{d}{d\left(t-t^{2}\right)}(t-t^{2})^{k}dt\\
& =1+\int_{0}^{1}\frac{1}{t}\left(t-t^{2}\right)\frac{d}{d\left(t-t^{2}\right)}\frac{\left(t-t^{2}\right)}{1-(t-t^{2})}dt\\
& =1+\int_{0}^{1}\frac{1}{t}\left(t-t^{2}\right)\frac{1}{\left((t-t^{2})-1\right)^{2}}dt\\
& =1+\int_{0}^{1}\frac{1-t}{\left(t^{2}-t+1\right)^{2}}dt\\
& =1+\int_{0}^{1}\frac{1-t}{\left(t^{2}-t+1\right)^{2}}dt\\
& =1+\int_{0}^{1}\frac{\frac{1}{2}-\left(t-\frac{1}{2}\right)}{\left(t-\frac{1}{2}\right)^{2}+\frac{3}{4}}dt\\
& =1-\frac{4}{9}\int_{-\frac{1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}}\frac{3x-\sqrt{3}}{\left(x^{2}+1\right)^{2}}dx\qquad,\qquad\frac{\sqrt{3}}{2}x=\left(t-\frac{1}{2}\right)\\
& =1-\frac{4}{9}\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\frac{3\tan u-\sqrt{3}}{\left(\tan^{2}u+1\right)^{2}}\frac{1}{\cos^{2}u}du\qquad,\qquad x=\tan u\\
& =1-\frac{4}{9}\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(3\sin u\cos u-\sqrt{3}\cos^{2}u\right)du\\
& =1-\frac{4}{9}\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\left(\frac{3}{2}\sin\left(2u\right)-\frac{\sqrt{3}}{2}\left(\cos\left(2u\right)+1\right)\right)du\\
& =1-\frac{4}{9}\left[-\frac{3}{4}\cos\left(2u\right)-\frac{\sqrt{3}}{4}\sin\left(2u\right)-\frac{\sqrt{3}}{2}u\right]_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\\
& =1-\frac{4}{9}\left[-\frac{\sqrt{3}}{2}\sin\left(2u\right)-\sqrt{3}u\right]_{0}^{\frac{\pi}{6}}\\
& =\frac{4}{3}+\frac{2\sqrt{3}\pi}{27}
\end{align*}
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ウォリス積分の定義
\[
\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta
\]
(*)log(1-x)のn乗の展開
\[
\log^{n}(1-x)=(-1)^{n}n!\sum_{k=0}^{\infty}\frac{S_{1}(k+n,n)}{(k+n)!}x^{k+n}
\]
ウォリス積分を含む極限
\[
\lim_{n\rightarrow\infty}\sqrt{n}\int_{0}^{\frac{\pi}{2}}\sin^{n}\theta d\theta=\sqrt{\frac{\pi}{2}}
\]
階乗と冪乗の極限
\[
\lim_{n\rightarrow\infty}\frac{x^{n}}{n!}=0
\]