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Question Number 168256 by mathlove last updated on 07/Apr/22

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{n!}{n^n}=?$$

Answered by MJS_new last updated on 07/Apr/22

$$\mathrm{first}\mathrm{thought}\frac{n!}{n^n}\mathrm{means}\frac{1\times 2\times ...\times n}{n\times n\times ...\times n}\mathrm{means}$$$$\frac{n-1numberssmallerthann}{n-1timesn}\mathrm{which}\mathrm{looks}$$$$\mathrm{a}\mathrm{lot}\mathrm{like}0$$$$$$$$\mathrm{prove}:$$$$n!\sim {\left(\frac{n}{\mathrm{e}}\right)}^n\sqrt{2\pi n}$$$$\frac{n!}{n^n}\sim \frac{\sqrt{2\pi n}}{{\mathrm{e}}^n}$$$$\mathrm{obviously}\mathrm{the}\mathrm{answer}\mathrm{is}0$$

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