Stirling's Theorem - 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):

\begin{align} n! &\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1 +{1\over12n}+{1\over288n^2} - {139\over51840n^3} -{571\over2488320n^4}+ \cdots \right) \\ &= \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \left(1+\frac{1}{(2^1)(6n)^1}+{1\over(2^3)(6n)^2}-{139\over(2^3)(2\cdot3\cdot5)(6n)^3} \right. \\ &\qquad\left. -{571\over(2^6)(2\cdot3\cdot5)(6n)^4} + \cdots \right). \end{align}

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 n → ∞, 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 shown in the next graph, which shows the relative error versus the number of terms in the series, for larger numbers of terms. More precisely, let S(n, t) 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:

\begin{align} \ln(n!) &\sim n\ln(n) - n + \tfrac{1}{2}\ln(2\pi n) +{1\over12n} -{1\over360n^3} +{1\over1260n^5} -{1\over 1680n^7} +\cdots \\ &= n\ln(n)-n+\tfrac{1}{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}\\ &\qquad - \frac{1}{(2^4 \cdot3^1 \cdot5^1\cdot7^1)n^7} +\cdots. \end{align}

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 Theorem

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

    “No speed of wind or water rushing by
    But you have speed far greater.”
    —Robert Frost (1874–1963)

    “Life is too short to waste
    In critic peep or cynic bark,
    Quarrel or reprimand:
    ‘Twill soon be dark;
    Up, heed thine own aim, and
    God speed the mark!”
    —Ralph Waldo Emerson (1803–1882)

    “‘Tis this desire of bending all things to our own purposes which turns them into confusion and is the chief source of every error in our lives.”
    —Sarah Fielding (1710–1768)

    “Writing a book I have found to be like building a house. A man forms a plan, and collects materials. He thinks he has enough to raise a large and stately edifice; but after he has arranged, compacted and polished, his work turns out to be a very small performance. The authour however like the builder, knows how much labour his work has cost him; and therefore estimates it at a higher rate than other people think it deserves,”
    —James Boswell (1740–1795)