How to Prove Stirling's Formula for ln(n!) ?

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SUMMARY

The discussion centers on proving Stirling's Formula for the natural logarithm of factorials, specifically ln(n!) = nln(n) - n + ln(2*pi*n)/2 for large n. Participants highlight that numerous resources online provide the proof, indicating its established nature in mathematical literature. The conversation emphasizes the formula's significance in asymptotic analysis and combinatorial mathematics.

PREREQUISITES
  • Understanding of asymptotic analysis
  • Familiarity with logarithmic properties
  • Basic knowledge of factorial functions
  • Concept of limits in calculus
NEXT STEPS
  • Study the derivation of Stirling's Formula in mathematical literature
  • Explore asymptotic expansions in combinatorial mathematics
  • Learn about the applications of Stirling's Formula in probability theory
  • Investigate the use of logarithms in analyzing growth rates
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Mathematics students, educators, and researchers interested in combinatorial analysis, asymptotic methods, and the properties of logarithmic functions.

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Homework Statement



Prove that ln(n!) = nln(n) - n + ln(2*pi*n)/2 for large n.


Homework Equations



I've been working on the proof but I just can't get it. Can anyone help me out?

The Attempt at a Solution

 
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You can find the proof at many places on the web.
 

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