Ball

The $n$-ball, denoted $\Bbb{B}^n$, is the interior of a Sphere $\Bbb{S}^{n-1}$, and sometimes also called the $n$-Disk. (Although physicists often use the term ``Sphere'' to mean the solid ball, mathematicians definitely do not!) Let ${\rm Vol}(\Bbb{B}^n)$ denote the volume of an $n$-D ball of Radius $r$. Then

$$\sum_{n=0}^\infty \mathop{\rm Vol}\nolimits (B^n) = e^{\pi r^2}[1+\mathop{\rm erf}\nolimits (r\sqrt{\pi}\,)],$$

where $\mathop{\rm erf}\nolimits (x)$ is the Erf function.

See also Alexander's Horned Sphere, Banach-Tarski Paradox, Bing's Theorem, Bishop's Inequality, Bounded, Disk, Hypersphere, Sphere, Wild Point

References

Freden, E. Problem 10207. ``Summing a Series of Volumes.'' Amer. Math. Monthly 100, 882, 1993.