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Question Number 173815 by mr W last updated on 18/Jul/22

prove (((n+1)/3))^n <n!

$${prove}\: \\ $$ $$\left(\frac{{n}+\mathrm{1}}{\mathrm{3}}\right)^{{n}} <{n}! \\ $$

Commented bymr W last updated on 19/Jul/22

good idea sir! thanks! n!≈(√(2πn))((n/e))^n lim_(n→∞) ((n+1)/( ((n!))^(1/n) )) =lim_(n→∞) (((n+1))/((2nπ)^(1/(2n)) ((n/e)))) =lim_(n→∞) (e/(((π/(1/(2n))))^(1/(2n)) ))(1+(1/n)) =lim_(n→∞) (e/(((π/(1/(2n))))^(1/(2n)) )) =lim_(x→0) (e/(((π/x))^x )) =(e/1) =e < 3

$${good}\:{idea}\:{sir}!\:{thanks}! \\ $$ $${n}!\approx\sqrt{\mathrm{2}\pi{n}}\left(\frac{{n}}{{e}}\right)^{{n}} \\ $$ $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}+\mathrm{1}}{\:\sqrt[{{n}}]{{n}!}} \\ $$ $$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\left({n}+\mathrm{1}\right)}{\left(\mathrm{2}{n}\pi\right)^{\frac{\mathrm{1}}{\mathrm{2}{n}}} \left(\frac{{n}}{{e}}\right)} \\ $$ $$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{e}}{\left(\frac{\pi}{\frac{\mathrm{1}}{\mathrm{2}{n}}}\right)^{\frac{\mathrm{1}}{\mathrm{2}{n}}} }\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right) \\ $$ $$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{e}}{\left(\frac{\pi}{\frac{\mathrm{1}}{\mathrm{2}{n}}}\right)^{\frac{\mathrm{1}}{\mathrm{2}{n}}} } \\ $$ $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}}{\left(\frac{\pi}{{x}}\right)^{{x}} } \\ $$ $$=\frac{{e}}{\mathrm{1}} \\ $$ $$={e}\:<\:\mathrm{3} \\ $$

Commented byFrix last updated on 18/Jul/22

⇔ 3>((n+1)/( ((n!))^(1/n) )) lim_(n→∞) ((n+1)/( ((n!))^(1/n) )) =e<3 just a idea...

$$\Leftrightarrow \\ $$ $$\mathrm{3}>\frac{{n}+\mathrm{1}}{\:\sqrt[{{n}}]{{n}!}} \\ $$ $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{n}+\mathrm{1}}{\:\sqrt[{{n}}]{{n}!}}\:=\mathrm{e}<\mathrm{3} \\ $$ $$\mathrm{just}\:\mathrm{a}\:\mathrm{idea}... \\ $$

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