$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} L_{p} & \equiv \lim_{n \to \infty}{\prod_{k = 1}^{n}\pars{1 + p/k} \over n^{p}} = \lim_{n \to \infty}\bracks{{1 \over n^{p}}\, {\prod_{k = 1}^{n}\pars{k + p} \over n!}} = \lim_{n \to \infty}\bracks{{1 \over n^{p}n!}\,\pars{1 + p}^{\large\overline{n}}} \\[5mm] & = \lim_{n \to \infty}\bracks{{1 \over n^{p}n!}\,{\Gamma\pars{1 + p + n} \over \Gamma\pars{1 + p}}} = {1 \over \Gamma\pars{1 + p}}\lim_{n \to \infty} \bracks{{1 \over n^{p}}\,{\pars{n + p}! \over n!}} \\[5mm] & = {1 \over \Gamma\pars{1 + p}}\,\lim_{n \to \infty}\bracks{{1 \over n^{p}}\, {\root{2\pi}\pars{p + n}^{p + n + 1/2}\expo{-p - n} \over \root{2\pi}n^{n + 1/2}\expo{-n}}}\quad\pars{\substack{Here,\ I\ use\ the\ well\ known\\[1mm] {\large z!\ Stirling\ Asymptotic\ Expansion} }} \\[5mm] & = {1 \over \Gamma\pars{1 + p}}\,\lim_{n \to \infty}\bracks{{1 \over n^{p}}\, {n^{n + p + 1/2}\,\pars{1 + p/n}^{3/2 + p + n}\,\expo{-p} \over n^{n + 1/2}}} \\[5mm] & = {1 \over \Gamma\pars{1 + p}}\,\lim_{n \to \infty} \bracks{\pars{1 + {p \over n}}^{n}\,\expo{-p}} = \bbx{1 \over \Gamma\pars{1 + p}} \end{align}
0 votos
$x!=\Gamma(x+1)$ y el Función gamma se define para los no enteros