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Find the maximum among the following :-

$1, \sqrt {2}, \sqrt[3]{3}... $

I think the answer is $\sqrt[3]{3}$.

Thanks for any help.

Soham
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    Even older http://math.stackexchange.com/questions/77935/prove-by-induction-that-for-all-n-geq-3-nn1-n1n/ – rtybase Sep 30 '16 at 17:14

1 Answers1

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By raising both sides to the $n(n+1)$-th power, it is enough to prove that $$ n^{n+1} > (n+1)^n $$ or, by dividing both sides by $n^n$, $$ n > \left(1+\frac{1}{n}\right)^n $$ that follows from $$ n > \color{red}{e} > \left(1+\frac{1}{n}\right)^n.$$

Jack D'Aurizio
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  • @Macavity: this answer was migrated from a duplicate question, only dealing with the $n>2$ case. In the actual case we obviously have to deal with $n=1$ and $n=2$ by hand. – Jack D'Aurizio Sep 30 '16 at 21:23