Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,406

James Stirling Formula

This is a brilliant approximation for factorials, particularly, factorials of higher order numbers.
For example, 1000! as per this formula is 4.023537292 x 10^2567. The actual value as per the calculator in the scientific mode is 4.0238726 x 10^2567.   :cool::cool:

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mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

sweetness!

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A logarithm is just a misspelled algorithm.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: James Stirling Formula

sweetness!

Very much so, if you think an error of

is ok.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

I just like how it contains both pi and e.

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A logarithm is just a misspelled algorithm.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: James Stirling Formula

sweetness!

Very much so, if you think an error of

is ok.

Which comes out to a relative error of 0.008%.  I'll take that.

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Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: James Stirling Formula

sweetness!

Very much so, if you think an error of

is ok.

.. not exactly big!

EDIT: Aah post collison

Last edited by Daniel123 (2007-12-11 05:37:01)

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

depends on what your priorities are i guess.

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A logarithm is just a misspelled algorithm.

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: James Stirling Formula

This is a brilliant approximation for factorials, particularly, factorials of higher order numbers.
For example, 1000! as per this formula is 4.023537292 x 10^2567. The actual value as per the calculator in the scientific mode is 4.0238726 x 10^2567.   :cool::cool:

So does this actually converge on the factorial value as n goes to infinity?

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: James Stirling Formula

well both n! and the approximation both diverge to infinity as n goes to infinity

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TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: James Stirling Formula

Yes.

http://en.wikipedia.org/wiki/Stirling%27s_approximation

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mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

i think he meant, does

Last edited by mikau (2007-12-11 06:36:33)

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A logarithm is just a misspelled algorithm.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: James Stirling Formula

I agree that percentage difference is more important than absolute difference.

If you consider the absolute argument the other way, you could say that 10mg of poison on your food is only 10mg more than the recommended amount and so not worth worrying about.

On the same theme, I would guess that this is false:

, but this is true:

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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: James Stirling Formula

isnt that identical? the only time that that would converge to 1, is if the first converged to 0?

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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: James Stirling Formula

Because the error may grow, but not as fast as n! grows.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

yeah. Note

but

Last edited by mikau (2007-12-11 13:20:49)

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A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: James Stirling Formula

What additional requirement can we impose so that Luca's statement holds?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: James Stirling Formula

Equality? That is, instead of

just getting arbitrarily close to 1, it actually has to get there.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: James Stirling Formula

Certainly you can come up with a restriction far less restricting than that.  Remember, this restriction can't apply to Mikau's example.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

if the two limits each converge to the same finite number?

Last edited by mikau (2007-12-12 05:08:17)

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A logarithm is just a misspelled algorithm.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: James Stirling Formula

Bingo.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: James Stirling Formula

It is also possible for the absolute error to approach 0 while the limits themselves diverge to infinity.  As a trivial example, let f(x) = x^2 and g(x) = x^2 + 1/x.  Then

and

but

Last edited by TheDude (2007-12-12 08:09:28)

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mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: James Stirling Formula

Bingo.

awesome! But are there any other restrictions that would do it?

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A logarithm is just a misspelled algorithm.