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 of, speed, error and/or estimates:

    “The terror of the atom age is not the violence of the new power but the speed of man’s adjustment to it—the speed of his acceptance.”
    —E.B. (Elwyn Brooks)

    “It was undoubtedly the feeling of exile—that sensation of a void within which never left us, that irrational longing to hark back to the past or else to speed up the march of time, and those keen shafts of memory that stung like fire.”
    —Albert Camus (1913–1960)

    “There is a lot of difference between tempting and leading into error. God tempts but does not lead into error. To tempt is to provide opportunities for us to do certain things if we do not love God, but putting us under no necessity to do so. To lead into error is to compel a man necessarily to conclude and follow a falsehood.”
    —Blaise Pascal (1623–1662)

    “A State, in idea, is the opposite of a Church. A State regards classes, and not individuals; and it estimates classes, not by internal merit, but external accidents, as property, birth, etc. But a church does the reverse of this, and disregards all external accidents, and looks at men as individual persons, allowing no gradations of ranks, but such as greater or less wisdom, learning, and holiness ought to confer. A Church is, therefore, in idea, the only pure democracy.”
    —Samuel Taylor Coleridge (1772–1834)