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Question Number 195259 by mathlove last updated on 28/Jul/23
prove that  lim_(n→∞)  ((e^n ∙(n!))/(n^n  (√n)))=(√(2π))
$${prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{e}^{{n}} \centerdot\left({n}!\right)}{{n}^{{n}} \:\sqrt{{n}}}=\sqrt{\mathrm{2}\pi} \\ $$
Commented by Frix last updated on 28/Jul/23
Easy because for n→∞: n!→(n^n /e^n )(√(2πn))
$$\mathrm{Easy}\:\mathrm{because}\:\mathrm{for}\:{n}\rightarrow\infty:\:{n}!\rightarrow\frac{{n}^{{n}} }{\mathrm{e}^{{n}} }\sqrt{\mathrm{2}\pi{n}} \\ $$

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