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*}ページ情報
| タイトル | 総和と総乗の逆順 |
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1のn乗根のべき乗の総和
\[
\sum_{k=0}^{n-1}\left(\omega_{n}^{\;k}\right)^{m}=n\delta_{0,\mod\left(m,n\right)}
\]
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\[
\sum_{k=1}^{\infty}\left|\alpha_{k}\right|<\infty
\]
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\[
\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k-1}}{2k-1}=\frac{\pi}{4}
\]
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\[
\prod_{k=1}^{\infty}\frac{\sqrt[2k-1]{\alpha}}{\sqrt[2k]{\alpha}}=2^{\Log\alpha}
\]