Volume of $n$-hyper-sphere derivation

Question

I am trying to prove the curse of dimensionality and on Wikipedia https://en.wikipedia.org/wiki/Curse_of_dimensionality
The volume of Hypersphere is given as
$\frac{2r^d \pi^{d/2}}{d\;\Gamma (d/2)}$
Can anybody help me that how we can derive this formula? as far as I know, the volume for $N$-dimensional hypersphere is $\frac{r^n\pi^{n/2}}{\;\Gamma (n/2+1)}$

Answer 1

The Gamma function has the property $\Gamma(z+1)=z\Gamma(z)$, so $$ \frac {r^{n}\pi ^{{n/2}}}{\Gamma (n/2+1)} = \frac {r^{n}\pi ^{{n/2}}}{\frac n2\Gamma (n/2)} = \frac {2r^{n}\pi ^{{n/2}}}{n\Gamma (n/2)}. $$ The two formulas you have therefore agree.

For more details on finding the volumes of balls and spheres, the Wikipedia links should get you started.