2重階乗の逆数和

2重階乗の逆数和

(1)

\[ \sum_{k=0}^{n}\frac{1}{\left(2k\right)!!}=\sqrt{e}\frac{\Gamma\left(n+1,\frac{1}{2}\right)}{\Gamma\left(n+1\right)} \]

(2)

\[ \sum_{k=0}^{\infty}\frac{1}{\left(2k\right)!!}=\sqrt{e} \]

(3)

\[ \sum_{k=0}^{n}\frac{1}{\left(2k+1\right)!!}=\sqrt{\frac{e\pi}{2}}\left(\frac{\Gamma\left(n+\frac{3}{2},\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}-1+\erf\left(\frac{\sqrt{2}}{2}\right)\right) \]

(4)

\[ \sum_{k=0}^{\infty}\frac{1}{\left(2k+1\right)!!}=\sqrt{\frac{e\pi}{2}}\erf\left(\frac{\sqrt{2}}{2}\right) \]

-

\(\Gamma\left(x\right)\)はガンマ関数、\(\Gamma\left(k,x\right)\)は第2種不完全ガンマ関数、\(n!!\)は2重階乗、\(\erf\left(x\right)\)は誤差関数

(1)

\begin{align*} \sum_{k=0}^{n}\frac{1}{\left(2k\right)!!} & =\sum_{k=0}^{n}\frac{1}{2^{k}k!}\\ & =\sum_{k=0}^{n}\frac{1}{k!}\left(\frac{1}{2}\right)^{k}\\ & =\sum_{k=+0}^{n}\sqrt{e}\left(\frac{\Gamma\left(k+1,\frac{1}{2}\right)}{\Gamma\left(k+1\right)}-\frac{\Gamma\left(k,\frac{1}{2}\right)}{\Gamma\left(k\right)}\right)\\ & =\sqrt{e}\left(\frac{\Gamma\left(n+1,\frac{1}{2}\right)}{\Gamma\left(n+1\right)}-\lim_{k\rightarrow0}\frac{\Gamma\left(k,\frac{1}{2}\right)}{\Gamma\left(k\right)}\right)\\ & =\sqrt{e}\frac{\Gamma\left(n+1,\frac{1}{2}\right)}{\Gamma\left(n+1\right)} \end{align*}

(2)

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

(2)-2

\begin{align*} \sum_{k=0}^{\infty}\frac{1}{\left(2k\right)!!} & =\sum_{k=0}^{\infty}\frac{1}{2^{k}k!}\\ & =\sum_{k=0}^{\infty}\frac{1}{k!}\left(\frac{1}{2}\right)^{k}\\ & =\sqrt{e} \end{align*}

(3)

\begin{align*} \sum_{k=0}^{n}\frac{1}{\left(2k+1\right)!!} & =\sqrt{\pi}\sum_{k=0}^{n}\frac{1}{2^{k+1}\left(k+\frac{1}{2}\right)!}\\ & =\sqrt{\frac{\pi}{2}}\sum_{k=0}^{n}\frac{1}{\left(k+\frac{1}{2}\right)!}\left(\frac{1}{2}\right)^{k+\frac{1}{2}}\\ & =\sqrt{\frac{\pi}{2}}\sum_{k=0}^{n}\sqrt{e}\left(\frac{\Gamma\left(k+\frac{1}{2}+1,\frac{1}{2}\right)}{\Gamma\left(k+\frac{1}{2}+1\right)}-\frac{\Gamma\left(k+\frac{1}{2},\frac{1}{2}\right)}{\Gamma\left(k+\frac{1}{2}\right)}\right)\\ & =\sqrt{\frac{e\pi}{2}}\left(\frac{\Gamma\left(n+\frac{3}{2},\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}-\frac{\Gamma\left(\frac{1}{2},\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)}\right)\\ & =\sqrt{\frac{e\pi}{2}}\left(\frac{\Gamma\left(n+\frac{3}{2},\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}-\frac{\Gamma\left(\frac{1}{2}\right)-\gamma\left(\frac{1}{2},\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)}\right)\\ & =\sqrt{\frac{e\pi}{2}}\left(\frac{\Gamma\left(n+\frac{3}{2},\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}-\frac{\Gamma\left(\frac{1}{2}\right)-\sqrt{\pi}\erf\left(\sqrt{\frac{1}{2}}\right)}{\Gamma\left(\frac{1}{2}\right)}\right)\\ & =\sqrt{\frac{e\pi}{2}}\left(\frac{\Gamma\left(n+\frac{3}{2},\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}-1+\erf\left(\frac{\sqrt{2}}{2}\right)\right) \end{align*}

(4)

\begin{align*} \sum_{k=0}^{\infty}\frac{1}{\left(2k+1\right)!!} & =\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{1}{\left(2k+1\right)!!}\\ & =\lim_{n\rightarrow\infty}\sqrt{\frac{e\pi}{2}}\left(\frac{\Gamma\left(n+\frac{3}{2},\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}-1+\erf\left(\frac{\sqrt{2}}{2}\right)\right)\\ & =\sqrt{\frac{e\pi}{2}}\erf\left(\frac{\sqrt{2}}{2}\right) \end{align*}

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2重階乗の逆数和

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