Maximum Ball

Geometry Level 2

What is the value of the dimension $n$ that maximizes the volume of unit $n$ -ball $\left\{ \left( x_1,x_2,\cdots,x_n \right) \in\mathbb{R}^n\, \mid \, x_1^2+x_2^2+\cdots+x_n^2\le 1 \right\}?$

Hint: The volume of $n$ -ball of radius $R$ is $V_n(R)=\frac{\pi^{\frac n2}}{\Gamma \left(\frac n2+1\right)}R^n,$ where $\Gamma(\cdot)$ denotes the gamma function.