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Does $\exists y >0$ , such that $\forall x>2$, $\sum_{z=1}^{\infty}( e^{-y(z-1/2)^x} -e^{-y(z)^x}) > 0.5$?

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By graphing the partial sums, the region $\sum_{z=1}^{\infty}( e^{-y(z-1/2)^x} -e^{-y(z)^x}) > 0.5$ when we set $y=0.87$, reaches arbitarily close to the $x=2$ line. Hence computationally sampling and applying Taylor's theorem in small intervals and bounding the tail of the sereis, I can prove that there exists a $y>0$ such that $\forall x>2.001$ the above holds. Is the above conjecture true for $x>2$, because by graphing the partial sums it seems to be true and if yes how can we try to attempt to prove it?


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