Well this one looks very interesting, it looks pretty obvious but I can't find a way to prove it formally:

Let a,b,c be positive numbers.

Prove:

$\displaystyle lim(a^n+b^n+c^n)^\frac{1}{n} = max(a,b,c)$

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- Nov 28th 2009, 02:50 AMadam63Limit of a sequence - nice one
Well this one looks very interesting, it looks pretty obvious but I can't find a way to prove it formally:

Let a,b,c be positive numbers.

Prove:

$\displaystyle lim(a^n+b^n+c^n)^\frac{1}{n} = max(a,b,c)$ - Nov 28th 2009, 03:16 AMBacterius
Try to define mathematically what $\displaystyle max(a, b, c)$ is (I believe it is the greatest of these numbers ?). You can't just leave a verbal operation in your limit, you must express it in algebraïc operations. It might be easier (Wondering)

- Nov 28th 2009, 03:44 AMPlato
- Nov 28th 2009, 04:40 AMchisigma
Let's suppose that $\displaystyle a>b$ and $\displaystyle a>c$ [othervise you can always swap the variables (Nod) ... ]. Because is...

$\displaystyle \sqrt[n]{a^{n} + b^{n} + c^{n}} = a \sqrt{1+(\frac{b}{a})^{n} + (\frac{c}{a})^{n}}$ (1)

... and...

$\displaystyle \frac{b}{a}<1$ , $\displaystyle \frac{c}{a}<1$ (2)

... from (1) and (2) we derive...

$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a^{n} + b^{n} + c^{n}} = a$ (3)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Nov 28th 2009, 04:43 AMadam63