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I was messing around and made some code to find the area of an n dimensional sphere. I noticed that as n increases, the area tends towards zero. These were the results:

area of 1 dimensional sphere: 2.0001751221664 area of 2 dimensional sphere: 3.1415926535898 area of 3 dimensional sphere: 4.189156979794 area of 4 dimensional sphere: 4.9348022005447 area of 5 dimensional sphere: 5.2642499169821 area of 6 dimensional sphere: 5.16771278005 area of 7 dimensional sphere: 4.7251796759576 area of 8 dimensional sphere: 4.0587121264168 area of 9 dimensional sphere: 3.2987977237512 area of 10 dimensional sphere: 2.5501640398773 area of 11 dimensional sphere: 1.8842688535664 area of 12 dimensional sphere: 1.3352627688546 area of 13 dimensional sphere: 0.91070849042345 area of 14 dimensional sphere: 0.59926452932079 area of 15 dimensional sphere: 0.38147668041015 area of 16 dimensional sphere: 0.23533063035889 area of 17 dimensional sphere: 0.14099345137557 area of 18 dimensional sphere: 0.082145886611128 area of 19 dimensional sphere: 0.046625683267975 area of 20 dimensional sphere: 0.025806891390014

The function to find the area of an n dimensional sphere is:

$$f(n) = \frac{\pi^{\frac{n}{2}}}{\Gamma{(1 + \frac{n}{2})}}$$

where n is the dimension. How would I calculate the sum of all n dimensional spheres?

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    $\begingroup$ Maple gives the answer as $-1+2e^\pi-e^\pi{\rm erfc}(\sqrt\pi)$. $\endgroup$ Commented Dec 19, 2014 at 2:40
  • $\begingroup$ Mathematica says the sum is $e^\pi(1+\mbox{erf}(\sqrt{\pi}))-1$, where erf$(z)$ is the error function $\frac{2}{\sqrt{\pi}}\int_0^z e^{-t^2}\,dt$. $\endgroup$ Commented Dec 19, 2014 at 2:42
  • $\begingroup$ use summation with n -> 0 to infinity. Question what do you mean by sum of all n dimensonal spheres? Area or what? $\endgroup$ Commented Dec 19, 2014 at 2:44
  • $\begingroup$ There seems to be something amiss---surely the volume of the $1$-ball should be exactly $2$. $\endgroup$ Commented Dec 19, 2014 at 2:44
  • $\begingroup$ what is a one dimensional sphere anyway? I thought spheres were 3d and circles 2d. This is going to mess with my mind $\endgroup$ Commented Dec 19, 2014 at 2:46

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