I just sent some time dicking around with the MacLaurin expansion of exp(-z(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}) to derive a series expression for √π, by integrating term-by-term along the real line. I'm not really concerned with wether this is a useful or well-studied expression, I just thought it would be a fun exercise.

Since the individual term-wise integrals diverge, we have to stick with a limit. It came down to

[itex]\frac{\sqrt{\pi}}{4}=\lim_{R\to\infty}\sqrt{R}\sum_{n=0}^{\infty}\frac{(-1)^nR^n}{(n-1)!}[/itex]

As R increases, the rate of convergence of the series decreases, so this is pretty useless unless we have a closed form for that series.

There's an obvious form in terms of the error function, but does anyone know of one that would actually lead to a nontrivial result?

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# Generalized geometric series

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