階乗を和に直しましょう

\begin{align*} \lim_{n\rightarrow\infty}\frac{1}{n}\sqrt[n]{\frac{\left(3n\right)!}{\left(2n\right)!}} & =\lim_{n\rightarrow\infty}\frac{1}{n}\sqrt[n]{Q\left(2n+1,n\right)}\\ & =\lim_{n\rightarrow\infty}\sqrt[n]{\frac{1}{n^{n}}\prod_{k=0}^{n-1}\left(2n+1+k\right)}\\ & =\lim_{n\rightarrow\infty}\sqrt[n]{\prod_{k=0}^{n-1}\left(2+\frac{1+k}{n}\right)}\\ & =\lim_{n\rightarrow\infty}\exp\left(\log\left(\prod_{k=0}^{n-1}\left(2+\frac{1+k}{n}\right)\right)^{\frac{1}{n}}\right)\\ & =\exp\left(\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\log\left(\left(2+\frac{1+k}{n}\right)\right)\right)\\ & =\exp\left(\int_{0}^{1}\log\left(2+x\right)dx\right)\\ & =\exp\left(\left[\left(2+x\right)\log\left(2+x\right)-\left(2+x\right)\right]_{0}^{1}\right)\\ & =\exp\left(3\log3-3-\left(2\log2-2\right)\right)\\ & =\exp\left(\log\frac{3^{3}}{2^{2}}-1\right)\\ & =\exp\left(\log\frac{3^{3}}{2^{2}e}\right)\\ & =\frac{27}{4e} \end{align*}