The $n^{th}$ root of the geometric mean of binomial coefficients. $\{{C_k^n}\}_{k=0}^n$ are binomial coefficients. $G_n$ is their geometrical mean. 
Prove 
$$\lim\limits_{n\to\infty}{G_n}^{1/n}=\sqrt{e}$$

$$\lim_{n\to\infty} G_n=\lim_{n\to\infty}\sqrt[n]{C_n^0C_n^1C_n^2\cdots C_n^n}=\lim_{n\to\infty}\sqrt[n]{\prod_{k=0}^n \binom{n}{k}}=\frac{n!}{G(n+2)^{2/n}},$$
where $G()$ is Barnes G-Function. $\lim_{n\to \infty}G_n$ diverges as can be seen here.
