総和と総乗の逆順
総和と総乗の逆順
(1)
\[ \sum_{k=a}^{b}f\left(k\right)=\sum_{k=-b}^{-a}f\left(-k\right) \](2)
\[ \prod_{k=a}^{b}f\left(k\right)=\prod_{k=-b}^{-a}f\left(-k\right) \](1)
\begin{align*} \sum_{k=a}^{b}f\left(k\right) & =f\left(a\right)+f\left(a+1\right)+\cdots+f\left(b\right)\\ & =f\left(b\right)+f\left(b+1\right)+\cdots+f\left(a\right)\\ & =f\left(-\left(-b\right)\right)+f\left(-\left(-b-1\right)\right)+\cdots+f\left(-\left(-a\right)\right)\\ & =\sum_{k=-b}^{-a}f\left(-k\right) \end{align*}(2)
\begin{align*} \prod_{k=a}^{b}f\left(k\right) & =\prod_{k=a}^{b}\exp\left(\Log\left(f\left(k\right)\right)\right)\\ & =\exp\left(\sum_{k=a}^{b}\Log\left(f\left(k\right)\right)\right)\\ & =\exp\left(\sum_{k=-b}^{-a}\Log\left(f\left(-k\right)\right)\right)\\ & =\sum_{k=-b}^{a}\exp\left(\Log\left(f\left(-k\right)\right)\right)\\ & =\prod_{k=-b}^{-a}f\left(-k\right) \end{align*}
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\prod_{k=1}^{\infty}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}}=2^{\Log\alpha}
\]
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\[
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\]
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\[
\sum_{k=1}^{n}\left(-1\right)^{k+1}=\frac{1}{2}+\frac{\left(-1\right)^{n+1}}{2}
\]
積の形の無限多重根号
\[
\sqrt[a_{1}]{r_{1}\sqrt[a_{2}]{r_{2}\cdots\sqrt[a_{n}]{r_{n}}}}=\exp\left\{ \sum_{k=1}^{n}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\}
\]