n!/√(2πn)e^n/n^n ~ 1 + O(1/n), whose infinite product over n blows up, giving an infinite figure of demerit. For convergence, we can improve Stirling's approximation: n! ~ √(2π(n+1/6)) n^n/e^n ~ n! (1+1/(144 n^2)+. . .). Then the infinite product of all the relative errors is In[86]:= Product[n!/√(2 π (n + 1/6)) n^n/E^n), {n, ∞}] Out[87]= E^( 2 Zeta'[-1] - 1/12) √(⅙! √(2π))) Out[88]= 1.00781097654253 (BtW, Stirling's contribution was "only" the √(2π); de Moivre had already found the rest. This must have been nearly as stunning as Euler's 𝜁(2) = π²/6.) Stirling's approximation can be improved no end. Perhaps the next better is E^-z √(2 π z) (1/(12 z) + z)^z ~ z! (1-1/(1440 z^3)+O[1/z]^4) whose "figure of merit" is Product[z!/(E^-z √(2 π z) (1/(12 z) + z)^z),{z,∞}] == E^(2 Zeta'[-1]) √√(2π)/(BarnesG[1 - I/(2√3)] BarnesG[1 + I/(2√3)]) ~ 1.001178221812, which seems to cry out for the BarnesG reflection formula. Which, unfortunately, only applies to G(z)/G(-z), not G(z)G(-z): E.g. BarnesG[1 - I/(2 √3)]/BarnesG[1 + I/(2 √3)] == (-1)^(1/24) E^(-((I PolyLog[2, E^(-(π/√3))])/(2 π))) (π Csch[π/(2 √3)])^(-(I/(2 √3))) —rwg