Stirling's approximation says that $$n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n.$$ What is known about constants $c_1$ and $c_2$ such that $$c_1\sqrt{n}\left(\dfrac{n}{e}\right)^n\le n!\le c_2\sqrt{n}\left(\dfrac{n}{e}\right)^n$$ for all $n$? (If we only want the statement for any sufficiently large $n$, then any $c_1<\sqrt{2\pi}<c_2$ should work.)
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1$\begingroup$ You might wanna have a look at this MathWorld@Wolfram site. $\endgroup$– Jose Arnaldo Bebita DrisOct 19, 2014 at 5:37
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$\begingroup$ Inequality (26) in particular should help a lot. $\endgroup$– Antonio VargasOct 19, 2014 at 15:13
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