How was Stirling's approximation derived?

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Stirling's approximation for factorials, expressed as x! ~ sqrt(2πx)(x/e)^x, is derived through mathematical analysis involving limits and integrals. The approximation provides a way to estimate large factorials efficiently. A detailed step-by-step derivation can be found in the article shared from Mathworld. This resource clarifies the underlying principles and calculations involved in the approximation. Understanding Stirling's approximation is essential for applications in statistics and combinatorics.
Paradox
I was wondering how Stirling's approximation

x! ~ sqrt(2[pi]x)xxe-x

was derived. Anyone know?
 
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This article at Mathworld derives Stirling's approximation step by step. :smile:
 
Thanks for the link, Jeff. It helped a lot :smile:
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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