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Prove that for $x_i > 0$ and $x_i$ distinct such that $x_1 < x_2... < x_{2n+1}$,

$\displaystyle\sum_{i = 1}^{2n + 1} (-1)^{i+1} x_i \leq \left(\sum_{i = 1}^{2n+1} (-1)^{i+1} (x_i)^n\right)^{\frac{1}{n}}$.

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    If we have $n=1$ and $x_1=2$ and $x_2=x_3=1$ we have on the left $1\cdot 2 -1+1$ and on the right side $(1\cdot 2 -1+1)^{\frac{1}{2}}$ which is less than the right hand side. If you take a closer look your equation says $f(n,x) \leq f(n,x)^{\frac1n} $ which does not hold in general2012-12-23
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    Sorry. $x_i$ should be ordered.2012-12-23
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    Induction? Yes?2012-12-23
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    I tried that but I got stuck.2012-12-23
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    You might have to formulate your induction hypothesis in a clever way-- see what different forms it can take on.2012-12-23

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