Proving stirlings formula


by Elysian
Tags: formula, proving, stirlings
Elysian
Elysian is offline
#1
Nov6-12, 07:48 PM
P: 33
1. The problem statement, all variables and given/known data

Prove that

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


2. Relevant equations


3. 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..
Phys.Org News Partner Science news on Phys.org
Better thermal-imaging lens from waste sulfur
Hackathon team's GoogolPlex gives Siri extra powers
Bright points in Sun's atmosphere mark patterns deep in its interior
Ray Vickson
Ray Vickson is online now
#2
Nov6-12, 08:48 PM
HW Helper
Thanks
P: 4,670
Quote Quote by Elysian View Post
1. The problem statement, all variables and given/known data

Prove that

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


2. Relevant equations


3. 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
Elysian
Elysian is offline
#3
Nov6-12, 10:50 PM
P: 33
Quote Quote by Ray Vickson View Post
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
MarneMath is offline
#4
Nov6-12, 11:07 PM
P: 422

Proving stirlings formula


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
Ray Vickson is online now
#5
Nov7-12, 01:27 AM
HW Helper
Thanks
P: 4,670
Quote Quote by Elysian View Post
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


Register to reply

Related Discussions
proving Stirling's formula.. help plz Introductory Physics Homework 6
Proving Rodrigues' formula Calculus 0
Help with Stirlings formula please!! General Math 4
proving a formula Calculus & Beyond Homework 4
Wallis Product into Stirlings Formula... General Math 3