- #1
Paradox
I was wondering how Stirling's approximation
x! ~ sqrt(2[pi]x)x^{x}e^{-x}
was derived. Anyone know?
x! ~ sqrt(2[pi]x)x^{x}e^{-x}
was derived. Anyone know?
Stirling's approximation is a mathematical formula used to approximate the value of factorials for large numbers. It is named after Scottish mathematician James Stirling.
Stirling's approximation uses a series expansion to approximate the factorial of a large number. It is based on the fact that as the number gets larger, the ratio of the factorial to the square root of the number approaches a constant value.
The formula for Stirling's approximation is n! ≈ √(2πn) * (n/e)^n, where n is the large number whose factorial is being approximated, π is the mathematical constant pi, and e is the mathematical constant Euler's number.
Stirling's approximation is used to calculate the approximate value of factorials for large numbers, which can be useful in various mathematical and scientific applications. It can also be used to estimate the values of other mathematical functions, such as the gamma function.
Stirling's approximation is only accurate for large values of n. For small values, it can produce significant errors. It also does not work well for values close to zero or negative values. Additionally, it is an approximation and not an exact value, so it may not be suitable for situations that require precise calculations.