Limit with Power Mean

If $p$ is a non-zero real number, and $x_1,x_2,\dots,x_n$ are positive real numbers, then the power mean with exponent $p$ of these positive real numbers is:

$M_p(x_1,x_2,\dots,x_n)=\left(\frac 1n\sum_{i=1}^n x_i^p\right)^{\frac 1p}.$

For $p=0$ we set it equal to the geometric mean (which is the limit of means with exponents approaching zero):

$M_0(x_1,x_2,\dots,x_n)=\sqrt[n]{\prod_{i=1}^n x_i}.$

Evaluate the limit below for all real number $p$ :

$\lim_{n\to\infty}\sqrt[n]{M_p\left(\binom n0,\binom n1,\dots,\binom nn\right)}.$

I got the incomplete answer (see problems Interesting Limit 24, 21 and wiki power mean inequality):

$\lim_{n\to\infty}\sqrt[n]{M_p\left(\binom n0,\binom n1,\dots,\binom nn\right)}= \begin{cases} 1,&\text{if }p\le -1\\ \sqrt e,&\text{if }p=0\\ 2,&\text{if }p\ge 1 \end{cases}$

Using Aaghaz Mahajan's result below, further answer is obtained:

$\lim_{n\to\infty}\sqrt[n]{M_p\left(\binom n0,\binom n1,\dots,\binom nn\right)}= \begin{cases} 1,&\text{if }p\le -1\\ \sqrt e,&\text{if }p=0\\ 2,&\text{if }p> 0 \end{cases}$

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Brian Lie Hello!! I think this result might help you for positive 'p'...........I have read it in a book, I can send you the book if you want.......

Aaghaz Mahajan - 2 years, 3 months ago

Dear Aaghaz, my email address is lixuepeili@gmail.com. Thank you for your sending.

Brian Lie - 2 years, 3 months ago

@Brian Lie Sir, could you suggest me any books to learn Dynamics from?? Or Physics in general??

Mechanics and Special Relativity by Freddie Williams is more mathematical.

Thanks Sir....!!

May I say something? I asked same question to Josh Silverman 3 years ago, he answered Kleppner and Klowenkow is the best.

Swapnil Das - 2 years, 3 months ago

Ohhhh!!! Thanks for the information!! I'll download it right away..... :)

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$