Stirling's Approximation

How it works

The factorial n! = 1·2·3·…·n grows faster than any polynomial and faster than any simple exponential. Stirling's approximation reveals its true growth rate: a power factor (n/e)ⁿ that dominates, multiplied by a square-root correction √(2πn) that smooths the answer to the right order. Once you accept the formula, the reason it works becomes intuitive — it is exactly the saddle-point approximation to the gamma-function integral.

The most useful form for analytical work is the logarithm:

log n! = n log n − n + ½ log(2πn) + 1/(12n) − 1/(360n³) + 1/(1260n⁵) − …

The first two terms n log n − n are the workhorse: that is the form used in every entropy calculation, every Boltzmann derivation, and every large-N combinatorial estimate. The ½ log(2πn) correction is enough for almost any numerical purpose. The 1/(12n) term is the first algebraic correction; beyond it the coefficients grow factorially and the series diverges, but the partial sums still give superb numerical approximations if truncated optimally.

Proof sketch — Laplace's method on Γ(n + 1)

Start from Euler's gamma integral n! = Γ(n + 1) = ∫₀^∞ tⁿ e^(−t) dt. Write the integrand as exp(n log t − t) and look for its maximum: differentiating n log t − t and setting to zero gives t = n. The integrand is sharply peaked there, so Taylor-expand the exponent around t = n with the substitution t = n(1 + u/√n):

n log t − t = n log n − n − u²/2 + O(u³/√n)

Plug this in, change variable, and the integral becomes:

n! ≈ nⁿ e^(−n) √n · ∫_{−∞}^∞ e^(−u²/2) du = nⁿ e^(−n) √n · √(2π)

which is exactly Stirling's formula. The √(2π) is the Gaussian normalization constant. Carrying higher-order terms in the Taylor expansion gives the 1/(12n) and subsequent corrections — the formal asymptotic series. This is the canonical example of Laplace's method or saddle-point integration in one dimension; the same technique gives asymptotics for Bessel functions, the central limit theorem, and a host of other integrals dominated by a sharp maximum.

Variants and refinements

• One-term form. n! ≈ (n/e)ⁿ — captures the dominant growth rate but is off by a factor of √(2πn), so the relative error grows with n. Useful for showing log n! ~ n log n − n, useless for numerics.
• Standard Stirling. n! ≈ √(2πn)(n/e)ⁿ. The default. Error 1/(12n) for the ratio, 1/(12n) for log n!.
• One-correction form. n! ≈ √(2πn)(n/e)ⁿ · (1 + 1/(12n)). Error 1/(288n²). Good enough for nearly anything; one extra multiplication.
• Two-correction form. n! ≈ √(2πn)(n/e)ⁿ · (1 + 1/(12n) + 1/(288n²) − 139/(51840n³)). Error ~ 1/n⁴. Used in libraries that need full double precision for moderate n.
• Nemes' form. n! ≈ √(2πn) · ((n + 1/(12n − 1/(10n)))/e)ⁿ. Closed form with similar accuracy to two-correction but only one expression.
• Ramanujan's form. n! ≈ √π · (n/e)ⁿ · (8n³ + 4n² + n + 1/30)^(1/6). Closed form, no division, error 10⁻⁵ at n = 5.
• Lanczos approximation. A rational approximation to Γ(z + 1) accurate to double precision for all moderate z. Used internally by most numerical libraries instead of plain Stirling because it converges, doesn't oscillate, and gives ≈ 15 correct digits with ≈ 10 terms.

Numerical examples

Computing the relative error of the leading Stirling formula for several values of n:

n          n!                          Stirling                    relative error
 1         1                           0.92214                     7.79%
 2         2                           1.91900                     4.05%
 5         120                         118.019                     1.65%
10         3,628,800                   3,598,696                   0.83%
20         2.43290 × 10¹⁸              2.42279 × 10¹⁸              0.416%
50         3.04141 × 10⁶⁴              3.03635 × 10⁶⁴              0.166%
100        9.33262 × 10¹⁵⁷             9.32485 × 10¹⁵⁷             0.0833%
1000       4.02387 × 10²⁵⁶⁷            4.02354 × 10²⁵⁶⁷            0.00833%

The relative error closely tracks 1/(12n), which is the leading term of the Stirling series. Adding the (1 + 1/(12n)) correction kills the 1/n term and leaves error ~ 1/(288n²) — at n = 10 the error drops from 0.83% to 0.0035%.

Entropy example: the multinomial 100! / (60! 30! 10!) counts ways to split 100 items into groups of sizes 60, 30, 10. Direct computation is fine here, but with Stirling:

log(100!/(60! 30! 10!)) ≈ −100 [0.6 log 0.6 + 0.3 log 0.3 + 0.1 log 0.1] − ½ log(2π·100·0.6·0.3·0.1)
                        ≈ 89.74 − 1.65 = 88.09 nats
(exact: 88.090…)

The Stirling estimate is correct to four significant figures and used as the leading term in Shannon's mutual information bounds, the Sanov large-deviation theorem, and free-energy expansions in statistical mechanics.

Common pitfalls

• Quoting it as exact. Stirling is an asymptotic approximation — the symbol is ≈ or ~, never =. At n = 1 it gives 0.92214, not 1. The error is small but never zero.
• Forgetting √(2πn). Many textbooks abbreviate to n! ≈ (n/e)ⁿ for the log form. That's fine when computing log n! − n log n + n (used in entropy), but if you need a numerical estimate of n!, you must include the √(2πn) — its omission is a factor-of-2.5 error at n = 10.
• Truncating the series too far. The Stirling series diverges. Adding terms past the smallest one makes the approximation worse. The optimal truncation is k* ≈ 2πn for given n; past that, the asymptotic series is no longer asymptotic.
• Using it for small n. The formula is meant for n ≫ 1. For n = 0 it gives nonsense (the formula doesn't define 0! = 1; you have to extend by the gamma function). For n = 1 the error is ~8%. For finite combinatorics with n ≤ 20, just multiply out the factorial.
• Confusing log n! with log Γ(n). n! = Γ(n + 1), so log n! and log Γ(n + 1) are equal. Some sources write Stirling for log Γ(z) which is the same formula with z replacing n + 1; an off-by-one shift trips up first-time readers.
• Misapplying in entropy of small samples. Sanov-type large-deviation bounds use Stirling to convert log(N!/Πnᵢ!) into −N Σ pᵢ log pᵢ. For small N the correction terms matter; ignoring them leads to over-confident estimates of rare-event probabilities.

Where Stirling shows up

• Combinatorics in the large-N limit. Binomial coefficient (n choose k) for k = αn has Stirling form ≈ 2^(nH(α)) where H is the binary entropy. Used in coding theory, random graphs, and Erdős-Ko-Rado style extremal arguments.
• Information theory. Shannon entropy of a discrete distribution is the Stirling limit of log multinomial / N. The asymptotic equipartition property, the Sanov large-deviation theorem, and rate-distortion theorems all use Stirling at their core.
• Statistical mechanics. Boltzmann entropy S = k log W counts microstates W; for an N-particle system, W ~ N! / Π Nᵢ!, and Stirling gives the thermodynamic limit S/N ≈ −Σ pᵢ log pᵢ. Same calculation derives the Boltzmann distribution by Lagrange multipliers, the Sackur-Tetrode formula for ideal-gas entropy, and the free energy of mixing.
• Quantum many-body systems. Partition functions for fermions and bosons involve products and sums that, in the thermodynamic limit, reduce to Gaussian integrals via Stirling. The Hartree-Fock and BCS mean-field theories use this.
• Random matrix theory. Wigner's semicircle law for the spectral density of large random matrices follows from a saddle-point evaluation of a determinantal expression that produces n! terms, simplified by Stirling.
• Number theory. The prime counting function π(x) ~ x / log x has a sharper form involving log of Stirling-style sums; the Selberg-Erdős elementary proof of the prime number theorem uses log n! asymptotics directly.
• Asymptotic analysis broadly. Stirling is the prototypical Laplace/saddle-point example. Any integral of the form ∫ e^(N S(x)) dx admits a similar Gaussian expansion around the saddle point of S — Stirling is the canonical worked example, and the techniques generalize to multi-dimensional and complex saddles.