TITLE: Find $\lim_{n\to\ \infty}\sqrt[n]{\frac{\sum_{i=1}^p a_i^n}{p}}$
QUESTION [3 upvotes]: I was trying to solve a question of an entrance exam. I am having trouble in the following problem. Please help me.
For positive real numbers $a_1, a_2, \ldots, a_p$ find the value of 
$$\displaystyle \lim_{n\to\ \infty}\sqrt[n]{\frac{\displaystyle \sum_{i=1}^p a_i^n}{p}}$$
What I have done so far:
From AM-GM inequality $\frac{\displaystyle \sum_{i=1}^p a_i^n}{p} \ge \sqrt[p]{\displaystyle \prod_{i=1}^p a_i^n} = \sqrt[\frac{n}{p}]{\displaystyle \prod_{i=1}^p a_i}$
So $\sqrt[n]{\frac{\displaystyle \sum_{i=1}^p a_i^n}{p}} \ge \sqrt[p]{\displaystyle \prod_{i=1}^p a_i}$
But then I can not find any way to proceed further. It will be very helpful for me any one provide me some help. I apologise for not showing much effort but I am really stuck.
Please help me. Thnx in advance.

REPLY [3 votes]: Set the vector $a=(a_1, a_2, \ldots, a_p)$.  Then $\|a\|_n=\sqrt[n]{\sum_{i=1}^p a_i^n}$, and the expression you have is $p^{-1/n}\|a\|_n$.  Taking $n\to \infty$ gives $p^0\|a\|_\infty=\|a\|_\infty$.    This is the $\infty$-norm, i.e. $$\|a\|_\infty=\max\{|a_1|,|a_2|,\ldots,|a_p|\}$$
[Note: this is a standard result in the theory of $p$-norms, see e.g. here].
{"set_name": "stack_exchange", "score": 3, "question_id": 1257485}