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# Difference between revisions of "Stirling's approximation"

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James Stirling (1692-1770, Scotland)

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

Because of [Euler-MacLaurin formula]

$\sum_{k=1}^N \ln k=\int_1^N \ln x dx+\sum_{k=1}^p\frac{B_{2k}}{2k(2k-1)}\left(\frac{1}{n^{2k-1}}-1\right)+R$

where B1 = −1/2, B2 = 1/6, B3 = 0, B4 = −1/30, B5 = 0, B6 = 1/42, B7 = 0, B8 = −1/30, ... are the Bernoulli numbers, and R is an error term which is normally small for suitable values of p.

$~\approx \int_1^N \ln x dx$

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

$~= N \ln N -N +1$

Thus, for large N

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