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Stirling's approximation

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James Stirling (1692-1770, Scotland)

\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N = \sum_{k=1}^N \ln k


~\approx \int_1^N \ln x dx


~= \left[ x \ln x - x \right]_1^N


~= N \ln N -N +1

Thus, for large N

\ln N! \approx  N \ln N -N