Perhaps @johncarlosbaez has some geometric intuition
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I was playing around, and stumbled upon something quite beautiful..
If you sum over the volumes of all 2k-dimensional unit spheres, that sum converges to e^π.
I have no idea why this should be true..pic.twitter.com/QvsQJkAuc2
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The general case of the sum over the volumes of all 2k-dimensional spheres of radius R
I have defined the dimensionless radius r to be the radius R measured in units of some "base length" R₀pic.twitter.com/SJjKy0o6Mr
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What does 2k sphere volume mean?
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it's just a sphere in 2k spatial dimensions
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Isn't it Maclaurin series formula? http://mathworld.wolfram.com/MaclaurinSeries.html …
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Yes, that’s how I arrived at the conclusion. But the fact that the sum is over all 2k-sphere volumes is the weird thing
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Does the sum over odd dimensional spheres also converge to something interesting?
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I think the answer is in the comments here
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Because denominator can be expressed as a gamma function, which will simplify to the exponent in your resulting term.
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I don’t see what geometric intuition that provides
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