Q1029 (Q1029): Difference between revisions

${\displaystyle {\sqrt[{n}]{\frac {(27)^{n}(n!)^{3}}{(3n)!}}}\sim \left({\frac {(27)^{n}\left({\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\right)^{3}}{{\sqrt {6\pi n}}\left({\frac {3n}{e}}\right)^{3n}}}\right)^{\frac {1}{n}}={\frac {27{\sqrt {2\pi }}^{\frac {3}{n}}n^{\frac {3}{2n}}\left({\frac {n}{e}}\right)^{3}}{{\sqrt {6\pi }}^{\frac {1}{n}}n^{\frac {1}{2n}}\left({\frac {3}{e}}\right)^{3}n^{3}}}=\left({\frac {2\pi n}{\sqrt {3}}}\right)^{\frac {1}{n}}}$