Prove that $\lim_{n \to \infty}(a^n + b^n+c^n) ^{(1/n)} = \max \{a,b,c\}$, for $a, b, c > 0$

Question

Prove that $\displaystyle\lim_{n \to \infty}\left(a^{n} + b^{n} +c^{n}\right)^{1/n} = \max\left\{a,b,c\right\}$ for $\displaystyle a, b, c > 0$.

as $p$ increases, $l^p$ norm converges to $l^\infty$ norm :) — mathworker21, Jun 09 '19 at 16:39

score 5 · Accepted Answer · answered Jun 09 '19 at 16:41

5

Estimate: $$ \max\{a^n,b^n,c^n\}\leq a^n+b^n+c^n\leq 3\max\{a^n,b^n,c^n\} $$ and so taking $n$-th root, $$ \max\{a,b,c\}\leq (a^n+b^n+c^n)^{1/n}\leq 3^{1/n}\max\{a,b,c\} $$ and squeeze.

answered Jun 09 '19 at 16:41

user10354138

33,887

Prove that $\lim_{n \to \infty}(a^n + b^n+c^n) ^{(1/n)} = \max \{a,b,c\}$, for $a, b, c > 0$

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