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Stirling Series

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The “classical” Stirling series is defined as (Dominic, 2008):
classical stirling series

Where the numbers Bk are the Bernoulli numbers.

The series was formulated by the French mathematician Abraham DeMoivre (1667-1994), based on work by Scottish mathematician James Stirling (1692-1770). Stirling wrote the series with powers of 1/(n + ½) (Gellinas, 2017).

Different Definitions for The Stirling Series

The Stirling series can be defined in several closely related ways. For example, as the asymptotic series for the gamma function:

n! = γ(n + 1) = ∫ x2 e-x dx (x: 0 → ∞).


Alternatively it can be defined as the asymptotic expansion of the factorial function n! (Angelis, 2009):
stirling series

The series is sometimes defined as an asymptotic expansion of Stirling’s formula, a good approximation for factorials:

log(n!) = n log n −n + ½ log(n) + log √ (2 π) + εn,

Where εn → 0 as n → ∞. It replaces epsilon(ε) with powers of 1/n (Conrad, 2020).

Stirling’s series can also be defined as the following divergent series (Impens, 2003)
stirling divergent

Where A, B, C… are positive constants.


References

Abramowitz, M. & Stegun, I. A. (Eds.) (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 257.
Angelis, V. (2009). Stirling’s Series Revisited. The American Mathematical Monthly
Vol. 116, No. 9 (Nov), pp. 839-843. Taylor & Francis, Ltd.
Arfken, G. (1985). “Stirling’s Series.” §10.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 555-559.
Conrad, K. (2020). Stirling’s Formula. Retrieved November 20, 2020 from https://kconrad.math.uconn.edu/blurbs/analysis/stirling.pdf
Dominici, D. (2008). Variations on a theme by James Stirling. Note di Matematica
Note Mat. 1.
Ferraro, G. (2008). The Rise and Development of the Theory of Series up to the Early 1820s. Springer.
Gelinas, J. (2017). Original proofs of Stirling’s series for log(n!). Retrieved November 20, 2020 from: https://arxiv.org/abs/1701.06689.
Impens, C. (2003). Stirling’s Series Made Easy. Retrieved November 20, 2020 from: https://cage.ugent.be/~ci/impens_stirling.pdf

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Stephanie Glen. "Stirling Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/stirling-series/
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