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 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:

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

Note by Brian Lie
1 month, 1 week ago

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Comments

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@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.......

- 1 month, 1 week ago

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

- 1 month, 1 week ago

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

- 1 month ago

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Mechanics and Special Relativity by Freddie Williams is more mathematical.

- 1 month ago

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Thanks Sir....!!

- 1 month ago

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

- 3 weeks, 6 days ago

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

- 3 weeks, 6 days ago

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