Limit of Hölder Mean as Exponent tends to Infinity

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Theorem

Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.

For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.


Then:

$\ds \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, \ldots, x_n} = \max \set {x_1, x_2, \ldots, x_n}$


Proof

Let $p \in \R$ such that $p \ne 0$.

Let it be assumed (or arranged) that:

$x_1 \ge x_2 \ge \cdots \ge x_n$

Then:

\(\ds \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, \ldots, x_n}\) \(=\) \(\ds \lim_{p \mathop \to +\infty} \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}\)
\(\ds \) \(=\) \(\ds \lim_{p \mathop \to +\infty} \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_1}^p \paren {\dfrac {x_k} {x_1} }^p}^{1 / p}\)
\(\ds \) \(=\) \(\ds \lim_{p \mathop \to +\infty} x_1 \paren {\frac 1 n \sum_{k \mathop = 1}^n \paren {\dfrac {x_k} {x_1} }^p}^{1 / p}\) taking $p$th root of ${x_1}^p$ and moving it outside the parenthesis
\(\ds \) \(=\) \(\ds x_1 \lim_{p \mathop \to +\infty} \paren {\frac 1 n \sum_{k \mathop = 1}^n \paren {\dfrac {x_k} {x_1} }^p}^{1 / p}\) Multiple Rule for Limits of Real Functions
\(\ds \) \(=\) \(\ds x_1\)
This article, or a section of it, needs explaining.
In particular: Somehow we equate $\ds \lim_{p \mathop \to +\infty} \paren {\frac 1 n \sum_{k \mathop = 1}^n \paren {\dfrac {x_k} {x_1} }^p}^{1 / p}$ to $1$, not sure how this is justified
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
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When this work has been completed, you may remove this instance of {{Explain}} from the code.
\(\ds \) \(=\) \(\ds \max \set {x_1, x_2, \ldots, x_n}\) by hypothesis: $x_1 \ge x_2 \ge \cdots \ge x_n$

$\blacksquare$


Sources

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