Stirling's approximation: Difference between revisions

Revision as of 11:56, 27 February 2007

James Stirling (1692-1770, Scotland)

\ln N! = \ln 1 + \ln 2 + \ln 3 + . . . + \ln N

= \sum_{k = 1}^{N} \ln k

\approx \int_{1}^{N} \ln x d x

= {[x \ln x - x]}_{1}^{N}

= N \ln N - N + 1

Thus, for large N

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

Revision as of 15:08, 21 February 2007 (edit) Carl McBride (talk \| contribs) m (New page: James Stirling (1692-1770, Scotland) :<math>\left.\ln N!\right. = \ln 1 + \ln 2 + \ln 3 + ... + \ln N</math> :<math> ~= \sum_{k=1}^N \ln k</math> :<math>~\approx \int_1^N \ln x dx</ma...)		Revision as of 11:56, 27 February 2007 (edit) (undo) Nice and Tidy (talk \| contribs) mNo edit summary Newer edit →
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	:<math>\ln N! \approx N \ln N -N</math>		:<math>\ln N! \approx N \ln N -N</math>
			[[Category: Mathematics]]