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# Difference between revisions of "Stirling's approximation"

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where ''B''<sub>1</sub> = −1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = −1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = −1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''. | where ''B''<sub>1</sub> = −1/2, ''B''<sub>2</sub> = 1/6, ''B''<sub>3</sub> = 0, ''B''<sub>4</sub> = −1/30, ''B''<sub>5</sub> = 0, ''B''<sub>6</sub> = 1/42, ''B''<sub>7</sub> = 0, ''B''<sub>8</sub> = −1/30, ... are the [http://en.wikipedia.org/wiki/Bernoulli_numbers Bernoulli numbers], and ''R'' is an error term which is normally small for suitable values of ''p''. | ||

− | Then | + | Then, for large ''N'', |

− | :<math>\ln N! \approx \int_1^N \ln x dx = | + | :<math>\ln N! \approx \int_1^N \ln x dx = N \ln N -N</math> |

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[[Category: Mathematics]] | [[Category: Mathematics]] |

## Revision as of 15:48, 28 March 2007

James Stirling (1692-1770, Scotland)

Because of Euler-MacLaurin formula

where *B*_{1} = −1/2, *B*_{2} = 1/6, *B*_{3} = 0, *B*_{4} = −1/30, *B*_{5} = 0, *B*_{6} = 1/42, *B*_{7} = 0, *B*_{8} = −1/30, ... are the Bernoulli numbers, and *R* is an error term which is normally small for suitable values of *p*.

Then, for large *N*,