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.

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