How Can Stirling's Formula Be Proven Using Calculus?

[Elysian]

Homework Statement

Prove that

lim_{n \rightarrow ∞} \frac{n! e^{n}}{n^{n+\frac{1}{2}}} = \sqrt{2π}

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/\sqrt{2π} 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..

 

[Ray Vickson]

Homework Statement

Prove that

lim_{n \rightarrow ∞} \frac{n! e^{n}}{n^{n+\frac{1}{2}}} = \sqrt{2π}

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/\sqrt{2π} 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

 

[Elysian]

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

 

[MarneMath]

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.

 

[Ray Vickson]

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