Heat and Thermodynamics

13.3 Stirling's Theorem

The value of n! increases very quickly with n. 30! produces a member requiring 33 figures to write. Stirling gave an approximate formula for n!

 

 

Taking log of both sides,

 

 

It can be proved in the following way

 

ln n! = ln1 + ln2 + ln3 + … + ln x + … + ln n

 

  ln nⁿ = ln n + ln n + ln n + … + ln n + … + ln n

 

or

 

 

The summation can be approximately replaced by the integration

 

 

or

 

 

or

 

ln n