We have N numbers $x_{k} \geq0 $. What is the $\lim _ { n \rightarrow \infty } \sqrt [ n ] { x _ { 1 } ^ { n } + \ldots + x _ { N } ^ { n } }$ ? Any hints?
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Show that it is $\max (x_1, \cdots, x_N)$. – Oct 11 '18 at 21:11
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As others have said, it is equal to $\text{max} (x_1,x_2,\dots,x_N)$. This is known as the infinity norm. – Oria Gruber Oct 11 '18 at 21:47
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WLOG Assume that $x_1=\max ${$x_1,x_2, \cdots x_k$}. Then we have $x_1 \le x_1+x_2+ \cdots +x_k \le kx_1$. Taking the limit we have:
$$x_1=\lim_{n \to \infty}{({x_1}^n)^{\frac{1}{n}}} \le \lim_{n \to \infty}{{({x_1}^n+{x_2}^n+ \cdots +{x_k}^n)}^{\frac {1}{n}} \le \lim_{n \to \infty}{(k {x_1}^n)^{\frac{1}{n}}}}=x_1$$ The result follows by the squeeze theorem.
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