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What is the value of the dimension nnn that maximizes the volume of unit nnn-ball {(x1,x2,⋯ ,xn)∈Rn ∣ x12+x22+⋯+xn2≤1}? \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\}?{(x1,x2,⋯,xn)∈Rn∣x12+x22+⋯+xn2≤1}?
Hint: The volume of nnn-ball of radius RRR is Vn(R)=πn2Γ(n2+1)Rn,V_n(R)=\frac{\pi^{\frac n2}}{\Gamma \left(\frac n2+1\right)}R^n,Vn(R)=Γ(2n+1)π2nRn, where Γ(⋅)\Gamma(\cdot)Γ(⋅) denotes the gamma function.
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