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- Feb 7, 2012

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So the challenge is to prove that\(\displaystyle \sum_{r=0}^{2n}(-1)^rx^{r^2} = 1-x + x^4 - x^9 + x^{16} -\ldots + x^{(2n)^2} > \frac12\) for all $x$ such that $0<x<1$ (outside that interval the result is obvious anyway).