カテゴリー: 式変形

\[ \sum_{k=1}^{m}\frac{1}{\left(\alpha k+\beta\right)^{n}}=\frac{\left(-1\right)^{n-1}}{\alpha^{n}\left(n-1\right)!}\left\{ \psi^{\left(n-1\right)}\left(m+1+\frac{\beta}{\alpha}\right)-\psi^{\left(n-1\right)}\left(1+\frac{\beta}{\alpha}\right)\right\} \]

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

\[ \sum_{m=a}^{\infty}\sum_{n=b}^{\infty}f(m,n)=\sum_{t=a+b}^{\infty}\sum_{s=a}^{t-b}f(s,t-s) \]

\[ \sum_{k=1}^{n}\frac{1}{\sqrt{k+\sqrt{k^{2}-1}}}=\frac{\sqrt{2}}{2}\left(\sqrt{n+1}+\sqrt{n}-1\right) \]