A curious limit

• Sep 2nd 2012, 10:28 PM
SworD
A curious limit
I stumbled upon this somewhere but didn't know how to prove it.

$\displaystyle \lim_{x\to\infty}\frac{e^x \cdot x!}{\sqrt{x} \cdot x^x}= \sqrt{2\pi}$

Anyone have any ideas why this is the case or how to derive it? I suspect it has something to do with the gamma function.
• Sep 2nd 2012, 10:47 PM
JJacquelin
Re: A curious limit
The Stirling's approximation of x! for large x is :
(x^x)*exp(-x)*sqrt((2x+(1/3))*pi) equivalent to (x^x)*exp(-x)*sqrt(2*pi)*sqrt(x)
• Sep 2nd 2012, 10:51 PM
SworD
Re: A curious limit
Ah, yeah when I look at that closely it works. Thanks
• Sep 2nd 2012, 10:55 PM
MaxJasper
Re: A curious limit
Quote:

Originally Posted by SworD
I stumbled upon this somewhere but didn't know how to prove it.

$\displaystyle \lim_{x\to\infty}\frac{e^x \cdot x!}{\sqrt{x} \cdot x^x}= \sqrt{2\pi}$

Anyone have any ideas why this is the case or how to derive it? I suspect it has something to do with the gamma function.

Expand the argument to its Taylor series and then let $\displaystyle x\to \infty$

$\displaystyle \lim_{x\to \infty } \, \frac{e^x x!}{\sqrt{x} x^x}$ = $\displaystyle \lim_{x\to\infty} \sqrt{2 \pi }+\frac{\sqrt{\frac{\pi }{2}}}{6 x}+\frac{\sqrt{\frac{\pi }{2}}}{144 x^2}-\frac{139 \sqrt{\frac{\pi }{2}}}{25920 x^3}-\frac{571 \sqrt{\frac{\pi }{2}}}{1244160 x^4}+\frac{163879 \sqrt{\frac{\pi }{2}}}{104509440 x^5}+O\left(\left(\frac{1}{x}\right)^6\right)$ = $\displaystyle \sqrt{2 \pi }$
• Sep 2nd 2012, 11:04 PM
SworD
Re: A curious limit
o.O Could you elaborate on how you arrived at that series? Unless you are extending JJacquelin's answer by inputting Stirling's formula then I'm confused.. and isnt this rather a Laurent series?