A new Stirling series as continued fraction

We introduce the following new Stirling series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $$\end{document} as a continued fraction, which is faster than the classical Stirling series.