# Stirling Series

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

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):

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)

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.