| Stirling gave a formula for estimating n! but here is better |
Presuming an offset c improving the approximation, e ~ (n+1/n)(n+c), we estimate by reduction considering a numerically advanced e ~ (n²/n²-1)(n²-1+c), pulling-out similar factors of e, = ((n/n-1)*(n/n+1))(n²-1+c) ~ e(n+1-c)*(n/n-1)(c²-c) * e(-n+c)*(n/n+1)(c²+c-1) which has factor(s) that would be unity, and zeroing-out the slowest, second term in (1-(1/n²))(c-c²)*(1+(1/n))(1-2c) ~ 1, whence c = 0.5, QED.
Alternatively, e ~ (n+.5/n-.5)n for aesthetic symmetry. [Or faster, (n+.5+(1/12n)/n-.5+(1/12n))n]
(We could have used L'Hospital's Rule on log of numerator and denominator going to infinity: lim (c-c²)(log n-1) + (1-c-c²)(log n+1) / (1-2c²)(log n) = 1; by derivation, lim (c-c²/n-1) + (1-c-c²/n+1) / (1-2c²/n) = 1; by design faster to 1, whence -1 + 2c = 0.)
Stirling's approximation to n! ~ (n/e)n*√2πn rewritten (n/e)(n+.5)*√2πe (which has a nice feature that there are no stray factors of n) yields n+1!/n+1 ~ (n+1/e)(n+.5)*√2πe-1 (which has a nice feature that it has an estimate for 0!)... the exponent has not changed: only baseline parameters n,√e have become n+1,√e-1, exactly as in our approximation of e (above). But midpoint peak (n+.5/e)(n+.5)*√2π (*) is yet-nearer ~ n!, asserted as follows ... (and also completely removes stray factors, and constants)
* (the slope of its curve, = 0 at c = 0: δc (n+.5-c/e)(n+.5)*√2π*ec = -(n+.5/e) + (n+.5-c/e) times a common positive: n ≠ -0.5)
[reconstruction ahead]
The comeback approximations to n = n!/n-1! are, by Stirling's rewritten (n/e)(n/n-1)(n-.5), the next higher (n/e)(n+1/n)(n+.5), and midpoint (n/e)√(2n+1/2n)(2n+.5)(2n/2n-1)(2n-.5)*(1-(1/4n²)).25, closer by reason of the square-root being a geometric average of doubly-higher order approximations to e; and the fourth-root being minuscule second-order compensation ~ (1/16n²) to even that (*). The initial 0! approximations are, Stirling's = 0, next higher, √2π/e ~ 0.922, and midpoint, √(π/e) ~ 1.075, which starts closest to unity,- and its successive factors, ×1×2×3×..., keep it closest:-- this is a "sterling approximation" to n! .
* (In fact -(1/48n²) is even closer-still than -(1/16n²) but hors d'oeuvre.)
[A faster expansion takes the baseline (n+.5) to (n+.5-(1/24(n+.5))) -- note that (n+.5) then does triple duty]
This also yields an approximation to n! for noninteger (and fractional) n .
(We can more finely estimate the value of fractional factorials by n! = n+k!/(n+k)(n+k-1)...(n+1) : k → N>>0) .
By the approximation, δ n! ~ (n+.5/e)(n+.5)*√2π*(log n+.5) , which comes back to n!*(log n+.5) ; And as (log n+.5)-(log .5) is the first approximation to the series sum, Σn (1/i) , thus δ:0! ~ (log .5) = -(log 2) . Cf the gamma function.
A premise discovery under the title,