整数と半整数の逆数和

整数と半整数の逆数和

(1)

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

(2)

\[ \sum_{k=0}^{\infty}\frac{1}{k!}=e \]

(3)

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

(4)

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

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\(\Gamma(x,y)\)は第2種不完全ガンマ関数、\(\erf(x)\)は誤差関数

(1)

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

(2)

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

(2)-2

\[ \sum_{k=0}^{\infty}\frac{1}{k!}=e \]

(3)

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

(4)

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

(4)-2

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

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整数と半整数の逆数和

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