Find $\lim_{n \to \infty}\left(\frac{(3n)!}{(2n)!n^n}\right)^{1/n}$

Question

Find $$\lim_{n \to \infty}\left(\frac{(3n)!}{(2n)!n^n}\right)^{1/n}$$

Answer 1

Given $$\lim_{n\rightarrow \infty}\bigg[\frac{(3n)!}{(2n)!\cdot n^n}\bigg]^{\frac{1}{n}}$$

Using stirling approximation $$n!=\bigg(\frac{n}{e}\bigg)^{\frac{1}{n}}\sqrt{2\pi n}$$

So $$\lim_{n\rightarrow \infty}\Bigg[\frac{\bigg(\frac{3n}{e}\bigg)^{3n}\sqrt{6\pi n}}{\bigg(\frac{2n}{e}\bigg)^{2n}\sqrt{4\pi n}\cdot n^n}\Bigg]^{\frac{1}{n}}.$$

So $$\lim_{n\rightarrow \infty}\frac{27}{4e}\bigg(\frac{\sqrt{6\pi n}}{\sqrt{2\pi n}}\bigg)^{\frac{1}{n}}=\frac{27}{4e}.$$