Proving stirlings formula

  • Thread starter Elysian
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In summary: Google is your friend.RGVGoogle is your friend.First, step, is look at the log(n!). Transform this into something useful and think about if it's a decreasing and increasing function. From there determine an inequality that is always true.Thanks but it doesn't really give me a decent method I can follow. Some of the methods make little sense to me. If this is homework, I'm assuming the short proof without a lot of detail is preferred?Yes, the proof can be brief if that is what you are looking for.
  • #1
Elysian
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Homework Statement



Prove that

lim[itex]_{n \rightarrow ∞}[/itex] [itex]\frac{n! e^{n}}{n^{n+\frac{1}{2}}}[/itex] = [itex]\sqrt{2π}[/itex]

Homework Equations

The Attempt at a Solution


Alright so for this problem I noticed it looked kind of similar to the integral formula for a normal distribution from statistics with the 1/[itex]\sqrt{2π}[/itex] in it, but I'm not really sure what to do. I imagine there's a sine and cosine somewhere in there but I'm not exactly sure how to bring it in, maybe via taylor polynomials through the terms given?

I've yet no definitive solution but I've got some basic outlines of ideas.. I'm right now in Calc 2 so i expect there to be series and sequences involved..
 
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  • #2
Elysian said:

Homework Statement



Prove that

lim[itex]_{n \rightarrow ∞}[/itex] [itex]\frac{n! e^{n}}{n^{n+\frac{1}{2}}}[/itex] = [itex]\sqrt{2π}[/itex]


Homework Equations




The Attempt at a Solution


Alright so for this problem I noticed it looked kind of similar to the integral formula for a normal distribution from statistics with the 1/[itex]\sqrt{2π}[/itex] in it, but I'm not really sure what to do. I imagine there's a sine and cosine somewhere in there but I'm not exactly sure how to bring it in, maybe via taylor polynomials through the terms given?

I've yet no definitive solution but I've got some basic outlines of ideas.. I'm right now in Calc 2 so i expect there to be series and sequences involved..

Google is your friend.

RGV
 
  • #3
Ray Vickson said:
Google is your friend.

RGV

Thanks but it doesn't really give me a decent method I can follow. Some of the methods make little sense to me
 
  • #4
My first question is this, how rigourous do you want this to be? It can be very long and in depth proof or it can be a paragraph. If this is homework, I'm assuming the short proof without a lot of detail is preferred?

First, step, is look at the log(n!). Transform this into something useful and think about if it's a decreasing and increasing function. From there determine an inequality that is always true.
 
Last edited:
  • #5
Elysian said:
Thanks but it doesn't really give me a decent method I can follow. Some of the methods make little sense to me

There are several web pages that present several approaches. If you don't like one of them , go to another. All of them require some calculus and some hard work.

RGV
 

1. What is Stirling's formula?

Stirling's formula is a mathematical approximation for the factorial of a large number. It states that n! is approximately equal to (√(2πn)) * (n/e)^n, where e is Euler's number (approximately 2.71828).

2. Who developed Stirling's formula?

The formula was developed by Scottish mathematician James Stirling in 1730.

3. How accurate is Stirling's formula?

Stirling's formula is an asymptotic approximation, meaning that it becomes more accurate as the input value becomes larger. For values greater than 10, it is accurate to six decimal places or better.

4. What are the applications of Stirling's formula?

Stirling's formula is commonly used in statistics, physics, and engineering to estimate the factorial of large numbers. It is also used in the analysis of algorithms and in the field of combinatorics.

5. How is Stirling's formula derived?

The formula can be derived using the method of steepest descent in complex analysis. It involves manipulating the gamma function, which is a generalization of the factorial function, to obtain the desired approximation.

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