What is the $\displaystyle \lim_{n\to \infty}\left[a^n+b^n+c^n\right]^{1/n}$, assuming that $0<a<b<c$?
I think that, as 1/n tends to zero, the limit 1. Is this correct?
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assuming that what？ – Zhang Yifeng Nov 21 '13 at 12:33
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@assume a,b,c > 0? – meta_warrior Nov 21 '13 at 12:36
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... assuming that $0\le a\le b\le c$, the answer would be $c$. – Hagen von Eitzen Nov 21 '13 at 12:38
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https://math.stackexchange.com/q/326172/321264 – StubbornAtom Apr 28 '20 at 07:55
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Hint: Assuming that $0\le a\le b\le c$, then
$$(c^n)^{\frac{1}{n}}\le(a^n+b^n+c^n)^{\frac{1}{n}}\le(3c^n)^{\frac{1}{n}}$$
Now let $n\to\infty$,...
meta_warrior
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@Marta: You need to use Squeeze theorem too... and the fact that $\lim_{n\to\infty}{3^{\frac{1}{n}}}=1$ – meta_warrior Nov 21 '13 at 13:00