Stirling approximation: relative error rapidly → 0
The relative error |n! − Stirling(n)| / n! falls below 1% at n = 8 and below 0.1% at n = 80. For large n, Stirling is essentially exact.
Stirling's formula: logarithmic form
ln(n!) ≈ n·ln(n) − n + ½·ln(2πn)
Equivalent: n! ≈ √(2πn) · (n/e)ⁿ
Relative error → 0 as n → ∞. Exact for all practical purposes when n ≥ 20.
log(n!) grows exactly as Stirling predicts
On a log scale, n! and Stirlings approximation are visually identical. Relative error approaches 0 as n grows.
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