Hölder Mean for Exponent -1 is Harmonic Mean

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:
 * $\map {M_{-1} } {x_1, x_2, \ldots, x_n} = \dfrac 1 {\dfrac 1 n \paren {\dfrac 1 {x_1} + \dfrac 1 {x_2} + \cdots + \dfrac 1 {x_n} } }$

which is the harmonic mean of $x_1, x_2, \ldots, x_n$.

Proof
Recall the definition of the Hölder mean with exponent $p$:
 * $\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$

Then:

which is the harmonic mean by definition.