Limit of (x! e^x) / (x^x *x^1/2)

1. Jul 28, 2005

roger

hi ,

1.)how do I find the limit of (x! e^x) / (x^x *x^1/2) as x tends to infinity ?

2.)and is f(x)= x! a function ? if so, how do I find the derivative ?

thanks for any help

Roger

2. Jul 28, 2005

robert Ihnot

Sterling's formula says limit $$n!=\frac{n^n}{e^n}\sqrt{2n\pi}$$ We need only substitute X for n, since the function is continuous for positive X, to work your problem.

3. Jul 28, 2005

George Jones

Staff Emeritus
I thought about saying this, but then I wondered whether the point of Roger's problem is to derive Stirling's formula.

Regards,
George

4. Jul 28, 2005

George Jones

Staff Emeritus
A standard extension of factorial to all real numbers except the negative integers is by way of the gamma function. Then $\Gamma (x + 1) = x!$. One way to derive Stirling's formula is by using the standard integral representation of the gamma function. A couple of the steps are, however, not completely obvious.

Regards,
George

5. Jul 29, 2005

roger

Can I find the limit without using sterlings formula ?

6. Jul 29, 2005

robert Ihnot

When I learned about Sterling's formula it was a graduate course and the professor put the derivation on the board. It is not that simple. Note the presence of $$\sqrt(2n\pi)$$. This frquently means the use of complex integration, but not here: http://courses.ncssm.edu/math/Stat_Inst/PDFS/appndx_1.pdf

Last edited: Jul 29, 2005
7. Jul 29, 2005

kant

interesting formula

8. Jul 30, 2005

roger

Thanks for the information.

Does the extension to real numbers excluding negative, mean that the factorial becomes 'continuous' so that a derivative exists ?

9. Jul 30, 2005

TenaliRaman

Yes and its usually expressed in terms of digamma function.

-- AI