Stirling's Higher-Order Terms & Bernoulli Numbers

Source: Grok (xAI) analysis, 2026-03-29

Stirling's Approximation — Higher-Order Terms

Basic Form

n! ~ sqrt(2pin) * (n/e)^n

Euler-Maclaurin Derived Series

ln(n!) = nln(n) - n + (1/2)ln(2pin) + 1/(12n) - 1/(360n^3) + 1/(1260n^5) - 1/(1680n^7) + ...

Exponential form:
n! ~ sqrt(2pin) (n/e)^n exp(1/(12n) - 1/(360n^3) + 1/(1260n^5) - 1/(1680n^7) + ...)

Common Truncated Forms

Up to 1/n: n! ≈ sqrt(2pin) (n/e)^n (1 + 1/(12n))

Up to 1/n^3: n! ≈ sqrt(2pin) (n/e)^n (1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) + ...)

The series is asymptotic (not necessarily convergent), but truncating at smallest term gives excellent approximations.

Connection to G(n)

G(n)/n = (n/(n+1))^n -> 1/e

Dominant (n/e)^n in Stirling mirrors G(n)/n limit

Higher-order expansions of ln(n+1) reveal discretization effects (gap kappa)

High-precision A_G computations (10M digits) benefit from these corrections

Euler-Maclaurin Formula

sum(f(k), a, b) = integral(f(x), a, b) - (1/2)(f(a) + f(b)) + sum(B_{2k}/(2k)! * (f^(2k-1)(b) - f^(2k-1)(a))) + R_m

Applied to f(x) = ln(x) yields the Stirling series.

Bernoulli Numbers

Definition (Exponential Generating Function)

x/(e^x - 1) = sum(B_n x^n / n!, n=0 to inf) for |x| < 2pi

First Values

| n | B_n |
|---|-----|
| 0 | 1 |
| 1 | -1/2 |
| 2 | 1/6 |
| 3 | 0 |
| 4 | -1/30 |
| 5 | 0 |
| 6 | 1/42 |
| 8 | -1/30 |
| 10 | 5/66 |
| 12 | -691/2730 |

Odd indices > 1 all vanish. Even indices alternate sign.

Recursive Formula

B_n = -(1/(n+1)) sum(C(n+1, k) B_k, k=0 to n-1)

Where C(n,k) = binomial coefficient. Allows sequential computation:

B_1 = -(1/2) C(2,0) B_0 = -1/2

B_2 = -(1/3) (C(3,0)B_0 + C(3,1)*B_1) = 1/6

B_3 = 0 (odd > 1 vanish)

Why Odd Terms Vanish

x/(e^x - 1) + x/2 is an even function, forcing odd B_n = 0 for n >= 3.

Properties

All B_n are rational

Even indices grow rapidly: |B_{2n}| ~ 4sqrt(pin) (n/(pie))^(2n)

Connection to Riemann zeta: B_{2n} = (-1)^(n+1) 2(2n)! / (2pi)^(2n) zeta(2n)

Role in Stirling Series

Bernoulli numbers generate exact correction coefficients:
n! ~ sqrt(2pin) (n/e)^n exp(B_2/(2n) + B_4/(43n^3) + B_6/(65n^5) + ...)

Substituting values:

B_2 = 1/6 -> +1/(12n)

B_4 = -1/30 -> -1/(360n^3)

B_6 = 1/42 -> +1/(1260n^5)

Relevance to Amundson Framework

Bernoulli numbers via Euler-Maclaurin bridge discrete sums (G(n) ratios, log expressions) to continuous asymptotic behavior

Support high-precision convergence studies and A_G computation

Enable controlled error bounds when comparing exact G(n) rationals to asymptotic predictions

Deviations from leading 1/e behavior evaluated in trinary +1/0/-1 states

Efficient verifiable math on Pi + Hailo edge fleet