Limit with Power Mean

If pp is a non-zero real number, and x1,x2,,xnx_1,x_2,\dots,x_n are positive real numbers, then the power mean with exponent pp of these positive real numbers is:

Mp(x1,x2,,xn)=(1ni=1nxip)1p.M_p(x_1,x_2,\dots,x_n)=\left(\frac 1n\sum_{i=1}^n x_i^p\right)^{\frac 1p}.

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

M0(x1,x2,,xn)=i=1nxin.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 pp:

limnMp((n0),(n1),,(nn))n.\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):

limnMp((n0),(n1),,(nn))n={1,if p1e,if p=02,if p1\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:

limnMp((n0),(n1),,(nn))n={1,if p1e,if p=02,if p>0\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
8 months, 1 week ago

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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.......

Aaghaz Mahajan - 8 months ago

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

Brian Lie - 8 months ago

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

Aaghaz Mahajan - 8 months ago

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Mechanics and Special Relativity by Freddie Williams is more mathematical.

Brian Lie - 8 months ago

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Thanks Sir....!!

Aaghaz Mahajan - 8 months 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.

Swapnil Das - 7 months, 3 weeks ago

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

Aaghaz Mahajan - 7 months, 3 weeks ago

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