MHB Can You Prove This Inequality Challenge?

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The inequality challenge focuses on proving that the series sum \( \sum_{r=0}^{2n}(-1)^rx^{r^2} = 1-x + x^4 - x^9 + x^{16} - \ldots + x^{(2n)^2} \) is greater than \( \frac{1}{2} \) for all \( x \) in the interval \( 0 < x < 1 \). Graphical analysis suggests that the function remains above \( \frac{1}{2} \) within this range. The discussion emphasizes the need for a formal proof of this assertion. Adding more terms to the series appears to reinforce the claim. The challenge invites mathematicians to explore and validate this inequality.
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In a recent https://mathhelpboards.com/threads/inequality-challenge.27634/#post-121156, anemone asked for a proof that $1-x + x^4 - x^9 + x^{16} - x^{25} + x^{36} > 0$. When I graphed that function, I noticed that in fact it is never less than $\frac12$. If you add more terms to the series, this becomes even more apparent:

https://www.physicsforums.com/attachments/311520._xfImport

So the challenge is to prove that$$ \sum_{r=0}^{2n}(-1)^rx^{r^2} = 1-x + x^4 - x^9 + x^{16} -\ldots + x^{(2n)^2} > \frac12$$ for all $x$ such that $0<x<1$ (outside that interval the result is obvious anyway).
 
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For $0<x<1$ the infinite series $$\sum_{r=0}^\infty (-1)^rx^{r^2}$$ is convergent. It is an alternating series, where the terms alternate in sign and decrease in absolute value. It has the property that the partial sums are alternately greater than, and less than, the sum of the whole series.

Let $$S = \sum_{r=0}^\infty (-1)^rx^{r^2}$$, and for $n\geqslant 1$ let $$S_n = \sum_{r=0}^n (-1)^rx^{r^2}$$. Then $S_{2n-1}<S<S_{2n}$. So to prove that $S_{2n} > \frac12$ it will be sufficient to show that $S\geqslant\frac12$.

To investigate $S$, use the Jacobi triple product, which says that $$\sum_{n=-\infty}^\infty x^{n^2}y^{2n} = \prod_{m=1}^\infty (1-x^{2m})(1+x^{2m-1}y^2)\left(1+\frac{x^{2m-1}}{y^2}\right)$$ for complex numbers $x,y$ with $|x|<1$ and $y\ne0$. For $0<x<1$ and $y=i$ that gives $$\sum_{n=-\infty}^\infty(-1)^n x^{n^2} = \prod_{m=1}^\infty (1-x^{2m})\bigl(1-(-1)^nx^{2m-1}\bigr)^2.$$ Each factor in that product is positive, so the infinite product must be positive or zero. Therefore $$\sum_{n=-\infty}^\infty(-1)^n x^{n^2} \geqslant 0.$$ But $$\sum_{n=-\infty}^\infty(-1)^n x^{n^2} = \sum_{n=-\infty}^0(-1)^n x^{n^2} + \sum_{n=0}^\infty(-1)^n x^{n^2} - 1$$ (the $-1$ coming from the fact that otherwise the $n=0$ term would be counted twice). Also, $(-1)^n x^{n^2}$ is the same for $-n$ as it is for $+n$. Therefore $2S-1\geqslant0$, so that $S\geqslant\frac12$, as required.
 
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