| Version current |
Version 4 |
| \PMlinkescapeword{fix} |
\PMlinkescapeword{fix} |
| \PMlinkescapeword{calculate} |
\PMlinkescapeword{calculate} |
| Fix $x_1, x_2, \ldots, x_n \in \mathbb{R}^+$. Then let |
Fix $x_1, x_2, \ldots, x_n \in \mathbb{R}^+$. Then let |
| \[ |
\[ |
| \mu(r) := \left(\frac{x_1^r+\cdots+x_n^r}{n}\right)^{1/r}. |
\mu(r) := \left(\frac{x_1^r+\cdots+x_n^r}{n}\right)^{1/r}. |
| \] |
\] |
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| For $r\neq 0$, by definition $\mu(r)$ is the $r$th power mean of the $x_i$. It is also clear that $\mu(r)$ is a differentiable function for $r\neq 0$. What is $\lim_{r\to 0} \mu(r)$? |
For $r\neq 0$, by definition $\mu(r)$ is the $r$th power mean of the $x_i$. It is also clear that $\mu(r)$ is a differentiable function for $r\neq 0$. What is $\lim_{r\to 0} \mu(r)$? |
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We will first calculate $\lim_{r\to 0} \log\mu(r)$ using \PMlinkname{l'H\^opital's rule}{LHpitalsRule}.
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We will first calculate $\lim_{r\to 0} \log\mu(r)$.
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| \begin{align*} |
\begin{align*} |
| \lim_{r\to 0} \log\mu(r) & = \lim_{r\to 0} \frac{\log\left(\frac{x_1^r+\cdots +x_n^r}{n}\right)}{r}\\ |
\lim_{r\to 0} \log\mu(r) & = \lim_{r\to 0} \frac{\log\left(\frac{x_1^r+\cdots +x_n^r}{n}\right)}{r}\\ |
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\intertext{Note that this is a limit of the form $0/0$ so we can apply \PMlinkname{l'H\^opital's rule.}{LHpitalsRule}} |
| & = \lim_{r\to 0} \frac{\left(\frac{x_1^r\log x_1+\cdots+x_n^r\log x_n}{n}\right)}{\left(\frac{x_1^r+\cdots+x_n^r}{n}\right)}\\ |
& = \lim_{r\to 0} \frac{\left(\frac{x_1^r\log x_1+\cdots+x_n^r\log x_n}{n}\right)}{\left(\frac{x_1^r+\cdots+x_n^r}{n}\right)}\\ |
| & = \lim_{r\to 0} \frac{x_1^r\log x_1+\cdots+x_n^r\log x_n}{x_1^r+\cdots+x_n^r}\\ |
& = \lim_{r\to 0} \frac{x_1^r\log x_1+\cdots+x_n^r\log x_n}{x_1^r+\cdots+x_n^r}\\ |
| & = \frac{\log x_1+\cdots+\log x_n}{n}\\ |
& = \frac{\log x_1+\cdots+\log x_n}{n}\\ |
| & = \log \sqrt[n]{x_1\cdots x_n}. |
& = \log \sqrt[n]{x_1\cdots x_n}. |
| \end{align*} |
\end{align*} |
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| It follows immediately that |
It follows immediately that |
| \[ |
\[ |
| \lim_{r\to 0} \left(\frac{x_1^r+\cdots+x_n^r}{n}\right)^{1/r} = \sqrt[n]{x_1\cdots x_n}. |
\lim_{r\to 0} \left(\frac{x_1^r+\cdots+x_n^r}{n}\right)^{1/r} = \sqrt[n]{x_1\cdots x_n}. |
| \] |
\] |
