Very Accurate Approximations for the Factorial Function

Abstract

We establish the following new Stirling-type approximation formulas for the factorial function n! approximate to root 2 pi n(n)e(-n) root n + 1/6 + 1/72n - 31/6480n(2) - 139/155520n(3) + 9871/6531840n(4) and n! approximate to root 2 pi n(n)e(-n) (4)root n(2) vertical bar n/3 vertical bar 1/18 2/405n 31/9270n(2). Our estimations give much more accurate values for the factorial function than some previously published strong formulas. We also derive new sequences converging to Euler-Mascheroni constant gamma very quickly.