カテゴリー: 総和・総乗

\[ \sum_{k=\left\lceil x\right\rceil }^{\left\lfloor y\right\rfloor }a_{k}b\left(k\right)=A\left(y\right)b\left(y\right)-\int_{x}^{y}A\left(t\right)b'\left(t\right)dt \]

\[ \sum_{k=1}^{\infty}\frac{\left(-1\right)^{k-1}}{2k-1}=\frac{\pi}{4} \]

\[ \sum_{k=0}^{n-1}\left(\omega_{n}^{\;k}\right)^{m}=n\delta_{0,\mod\left(m,n\right)} \]

\[ \sum_{k=a}^{b}\frac{f\left(k\right)}{f\left(k\right)+f\left(a+b-k\right)}=\frac{b-a+1}{2} \]

\[ \sum_{k=a}^{b}f\left(k\right)=\sum_{k=a}^{b}f\left(a+b-k\right) \]

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

\[ \sum_{k=a}^{b}f\left(k\right)=\sum_{k=-b}^{-a}f\left(-k\right) \]

\[ \sum_{k=1}^{\infty}\left|\alpha_{k}\right|<\infty \]

\[ 1\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3 \]

\[ \prod_{k=1}^{\infty}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}}=2^{\Log\alpha} \]

\[ \sqrt[a_{1}]{r_{1}\sqrt[a_{2}]{r_{2}\cdots\sqrt[a_{n}]{r_{n}}}}=\exp\left\{ \sum_{k=1}^{n}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\} \]