The sum of the volumes of an infinite number of unit $n$-spheres each increasing in dimension is given by $$V_{\infty}= \sum_{n=0}^{\infty} V(n)$$ which gives us $$\sum_{n=0}^{\infty}\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)} = 46 \, \,(- 0.00067...)$$ which is surprisingly close to an integer. Its closed form can be written as $$V_{\infty} = e^{\pi}(\text{erf}(\sqrt{\pi})+1)$$
For the surface area equivalent, its closed form is $$S_{\infty} = 2\pi V_{\infty}+2$$ $$= 289 \, \,(+ 0.22289...)$$ For $S_{\infty}$, though it has a nice closed form, I wouldn't consider it being 'strangely' close to an integer. However I wonder why $V_{\infty}$ is so close to an integer value from a philosophical point of view as well as mathematical. It also seems like its closed form is weirdly 'nice' containing $e^{\pi}$ and $\sqrt{\pi}$ as argument of the error function, so for that to be this close to an integer could perhaps mean it is less likely to be a coincidence but this reasoning isn't mathematical so it doesn't really matter how 'nice' the closed form is. I'm curious, and look forward to know what you guys think
