Another question regarding spheres:
Is it true that $|S_{n}| = O( \frac{1}{\sqrt n} )$ ?
(I mean the surface area of the n dimensional sphere) Thanks !
Another question regarding spheres:
Is it true that $|S_{n}| = O( \frac{1}{\sqrt n} )$ ?
(I mean the surface area of the n dimensional sphere) Thanks !
Use Stirling's approximation $\Gamma(\frac n2)=(\frac n2 -1)!\approx (n/2)^{n/2}e^{-n/2}$, so $ S_n=\frac{2\pi^{n/2}}{\Gamma(\frac12 n)} \approx\frac{2\pi^{n/2}}{(\frac n2-1)^{\frac n2-1}e^{-\frac n2+1}} =2\left(\frac{\pi e^{1-2/n}}{(n/2-1)^{1-2/n}}\right)^{n/2}\\ \approx \left(\frac1n\right)^{n/2} =\left(\frac1{\sqrt{n}} \right)^{n} < \left(\frac1{\sqrt{n}} \right), $ for $n>1$ (then $(n/2-1)^{1-2/n}\sim (n/2)\;$ for large $n$). So $S_n$ doesn't grow faster than $O\left(\frac1{\sqrt{n}}\right)$.