Stirling's Approximation - Speed of Convergence and Error Estimates

Speed of Convergence and Error Estimates

More precisely,

with

Stirling's formula is in fact the first approximation to the following series (now called the Stirling series):

$n!\sim\sqrt{2\pi n}\left({n\over e}\right)^n \left( 1 +{1\over12n} +{1\over288n^2} -{139\over51840n^3} -{571\over2488320n^4} + \cdots \right)$

$=\sqrt{2\pi n}\left({n\over e}\right)^n \left( 1 +{1\over(2^1)(6n)^1} +{1\over(2^3)(6n)^2} -{139\over(2^3)(2\cdot3\cdot5)(6n)^3} -{571\over(2^6)(2\cdot3\cdot5)(6n)^4} + \cdots \right).$

Explicit formula for the coefficients in this series was given by G. Nemes . The first graph in this section shows the relative error vs. n, for 1 through all 5 terms listed above.

As, the error in the truncated series is asymptotically equal to the first omitted term. This is an example of an asymptotic expansion. It is not a convergent series; for any particular value of n there are only so many terms of the series that improve accuracy, after which point accuracy actually gets worse. This is demonstrated in the next graph, which shows the relative error vs. the number of terms in the series, for larger numbers of terms. (More precisely, let be the Stirling series to t terms evaluated at n. The graphs show, which, when small, is essentially the relative error.)

Writing Stirling's series in the form:

$\ln n!\sim n\ln n - n + {1\over 2}\ln(2\pi n) +{1\over12n} -{1\over360n^3} +{1\over1260n^5} -{1\over 1680n^7} +\cdots$

$=n\ln n - n + {1\over 2}\ln(2\pi n) +{1\over(2^2\cdot3^1)n} -{1\over(2^3\cdot3^2\cdot5^1)n^3} +{1\over(2^2\cdot3^2\cdot5^1\cdot7^1)n^5} -{1\over(2^4\cdot3^1\cdot5^1\cdot7^1)n^7} +\cdots.$

it is known that the error in truncating the series is always of the same sign and at most the same magnitude as the first omitted term.

Read more about this topic: Stirling's Approximation

Famous quotes containing the words speed, error and/or estimates:

“Among the laws controlling human societies there is one more precise and clearer, it seems to me, than all the others. If men are to remain civilized or to become civilized, the art of association must develop and improve among them at the same speed as equality of conditions spreads.”
—Alexis de Tocqueville (1805–1859)

“Error is a supposition that pleasure and pain, that intelligence, substance, life, are existent in matter. Error is neither Mind nor one of Mind’s faculties. Error is the contradiction of Truth. Error is a belief without understanding. Error is unreal because untrue. It is that which seemeth to be and is not. If error were true, its truth would be error, and we should have a self-evident absurdity—namely, erroneous truth. Thus we should continue to lose the standard of Truth.”
—Mary Baker Eddy (1821–1910)

“And, by the way, who estimates the value of the crop which nature yields in the still wilder fields unimproved by man? The crop of English hay is carefully weighed, the moisture calculated, the silicates and the potash; but in all dells and pond-holes in the woods and pastures and swamps grows a rich and various crop only unreaped by man.”
—Henry David Thoreau (1817–1862)