Sum of all Spheres

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

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

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

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 while back. This function is defined as

and reduces to the exponential function when I = I^2 = 1.

The sum of the volumes of all unit spheres of all dimensions is E_{1/2, 1}(aI). And from the post mentioned above,

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

Now erfc(-aI) a 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 E_{3/2, 1}(I^3/2).

• Sphere packing
• Homotopy groups of spheres
• Integrating polynomials over a sphere