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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総和と総乗の逆順

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