How to proof stirling approximation

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SUMMARY

The discussion centers on Stirling's approximation, specifically the expression $$\ln n! \approx n \ln n - n$$. A participant clarifies that while this approximation is widely accepted, it is crucial to understand that it is not an exact equality. The approximation is derived from the integral of the logarithm function, which provides a method to estimate the factorial of large numbers. The discussion emphasizes the importance of recognizing the approximation's nature and the existence of formal proofs and error bounds available on Wikipedia.

PREREQUISITES
  • Understanding of logarithmic functions
  • Familiarity with factorial notation
  • Basic knowledge of calculus, specifically integration
  • Concept of approximations in mathematical analysis
NEXT STEPS
  • Study the formal proofs of Stirling's approximation available on Wikipedia
  • Explore error bounds associated with Stirling's approximation
  • Learn about the integral approximation techniques in calculus
  • Investigate other mathematical approximations and their applications
USEFUL FOR

Mathematicians, students studying calculus, and anyone interested in numerical methods for approximating factorials will benefit from this discussion.

Another1
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View attachment 8757

i want to know about stirling approximation. why $$lnx! = xlnx - x$$
 

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Another said:
i want to know about stirling approximation. why $$lnx! = xlnx - x$$

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.
 
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.
 

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