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The sum is a nice corollary, but not what I'd seek to explain. First, it's rare that one should add a length, an area, a volume... it goes against dimensional analysis. Second, e^pi is not a number I know anything about.
I agree, but I was just thinking we could attach a dimensional factor (1/L)^n if we wanted too which would leave the exponential in tact
The factor could be taken as some "base" volume
Hmm, if you do that, can you interpret the result as some weird Riemannian manifold, and the sum as the exponential map from a particular special point? (Maybe the origin?)
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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..
