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I am trying to evaluate the Casimir force using a Gaussian regulator (I know there are other much easier ways to do this, but I want to try this!) We then are reduced to evaluating the sum

$$ \sum\limits_{n=1}^\infty n e^{-\alpha n^2} $$

Moreover, I am interested in the series expansion of the above sum around $\alpha = 0$. Any ideas how I would go about obtaining this sum?

PS - I don't want to use the Euler-Mclaurin formula.That was used to show that a general regulator would always give one the same answer, so this would just be a special case of the that proof and not a very novel way. Any other ideas?

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Well, you can just use the Taylor expansion of the exponential $$\sum_{n=1}^\infty n e^{-\alpha n^2}=\sum_{n=1}^\infty n \sum_m \frac{(-\alpha n^2)^m}{m!}$$ and use the definition of the Riemann zeta function $$\zeta (s)=\sum_{n=0}^\infty\frac{1}{n^s}$$ so you have $$\sum_m \frac{(-\alpha)^{m})\zeta(-2m-1)}{m!}$$ and finally use the zeta function regularization.

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  • $\begingroup$ Hi JAAT, welcome to physics.SE! This looks promising, but I'm not sure it's going to work. Can you elaborate? Thanks! $\endgroup$ – AccidentalFourierTransform Dec 4 '18 at 22:56

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