Sums of volumes of spheres

I ran across a video this afternoon that explains that the sum of volumes of all even-dimensional unit spheres equals eπ.

Why is that? Define vol(n) to be the volume of the unit sphere in dimension n. Then

\mathrm{vol}(n) = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)}

and so the sum of the volumes of all even dimensional spheres is

\sum_{k=0}^\infty \mathrm{vol}(2k) = \sum_{k=0}^\infty \frac{\pi^k}{k!} = \exp(\pi)

But what if you wanted to sum the volumes of all odd dimensional unit spheres? Or all dimensions, even and odd?

The answers to all these questions are easy in terms of the Mittag-Leffler function that I blogged about a couple years ago. This function is defined as

E_{\alpha, \beta}(x) = \sum_{k=0}^\infty \frac{x^k}{\Gamma(\alpha k+\beta)}

and reduces the exponential function when α = β = 1.

The sum of the volumes of all unit spheres of all dimensions is E1/2, 1(√π). And from the post mentioned above,

E_{1/2, 1}(x) = \exp(x^2) \, \mbox{erfc}(-x)

where erfc is the complementary error function. So the sum of all volumes of spheres is exp(π) erfc(-√π).

Now erfc(-√π) ≈ 1.9878 and so this says the sum of the volumes of spheres of all dimensions is about twice the sum of the even dimensional spheres alone. And so the sum of the odd dimensional unit sphere volumes is almost the same as the sum of the even dimensional ones.

By the way, you could easily answer other questions about sums of sphere volumes in terms of the Mittag-Leffler function. For example, if you want to add up the volumes in all dimensions that are a multiple of 3, you get E3/2, 13/2).

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