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?