\displaystyle\lim_{n\to\infty}\frac{e^{n/4}\cdot\sqrt[n]{1^1\cdot 2^2\cdot\ldots\cdot n^n}}{n^{(n+1)/2}} = \\
\lim_{n\to\infty}\frac{e^{n/4}\cdot\sqrt[n]{n^1\cdot n^2\cdot\ldots\cdot n^n}\sqrt[n]{(\frac{1}{n})^1\cdot (\frac{2}{n})^2\cdot\ldots\cdot (\frac{n}{n})^n}}{n^{(n+1)/2}}