How to proof stirling approximation

[Another1]

View attachment 8757

i want to know about stirling approximation. why \(\displaystyle lnx! = xlnx - x\)

 

[I like Serena]

i want to know about stirling approximation. why \(\displaystyle lnx! = xlnx - x\)

Hi Another!

Wiki explains Stirling's approximation.

We can see that it's true because:
$$\ln n! = \sum_{k=1}^n \ln k \approx \int_1^n \ln x\,dx = (x\ln x - x)\Big|_1^n = n\ln n - n + 1$$
The wiki page has formal proofs and bounds on the error.

 

[HOI]

First, do you understand that "Stirling's Approximation" is an approximation. There is NO proof that "ln(x!)= xln(x)- x" because that is NOT true- they are approximately equal, not equal.