[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the area of the unit sphere in [math]\mathbb R^N[/math] is given by

[[math]] A=\left(\frac{\pi}{2}\right)^{[N/2]}\frac{2^N}{(N-1)!!} [[/math]]

with our usual convention [math]N!!=(N-1)(N-3)(N-5)\ldots[/math] for double factorials.