TITLE: Find $L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$
QUESTION [5 upvotes]: Find the limit following:
$$L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$$
With $x,\: y\: z\in R$
P.S
I think this limit result is $L=max\left\{x,\: y\: z \right\}$. But i'm not find it, so expect people to help me find out the results by some solution
Extend this limit:
$$S=\lim \limits_{n\to \infty}\sqrt[n]{\sum_{i}^{m}a_{i}}$$ with $a_i\in R,\: i=1,\: m$

REPLY [15 votes]: Put $M = \max(x,y,z).$  Then you have $M^n \le x^n + y^n + z^n \le 3M^n$.  Take $n$th roots to ge
$$M \le \root{n}\of{x^n + y^n + z^n} \le 3^{1/n}M.$$
Let $n\to \infty$ and see your limit is $M$.

REPLY [2 votes]: The answer should be the max over all terms.
Think of the term which will start dominating the sum when n becomes arbitrarily large ...