https://en.wikipedia.org/wiki/Stirling%27s_approximation#A_convergent_versio... 1) DeMoivre had it first, but didn't recognize the √2π. 2) There's a convergent series (whose?) for ln x!, corresponding to the usual divergent In[837]:= BernoulliB[n + 1] ----------------- n ----------------- n + 1 Table[-----------------, {n, 11}] n x 1 1 1 1 {----, 0, -(------), 0, -------, 0, -(-------), 0, 12 x 3 5 7 360 x 1260 x 1680 x 1 691 -------, 0, -(----------)} 9 11 1188 x 360360 x (Divergent because Bernoulli_n, like n!, outgrows x^n for any x.) But with "rising factorials" (Pochhammers) instead of x^n, and weighted row sums of Stirling's Triangle instead of Bernoullis, Out[839]= 1 1 ---------- + ------------------ + 12 (1 + x) 12 (1 + x) (2 + x) 59 --------------------------- + 360 (1 + x) (2 + x) (3 + x) 29 ---------------------------------- + 60 (1 + x) (2 + x) (3 + x) (4 + x) 533 ------------------------------------------- + 280 (1 + x) (2 + x) (3 + x) (4 + x) (5 + x) 1577 --------------------------------------------------- 168 (1 + x) (2 + x) (3 + x) (4 + x) (5 + x) (6 + x) + 280361 / (5040 (1 + x) (2 + x) (3 + x) (4 + x) (5 + x) (6 + x) (7 + x)) Both of these series get added to the usual Stirling approximation ln(x^x/e^x √(2πx)). The coefficients of this convergent series actually outgrow the Bernoullis, but the Pochhammers grow faster still. ("Pochhammers" is so appealing compared to "rising factorial powers". Too bad that, as usual, Pochhammer didn't invent them.) Expanding the convergent series termwise about ∞, In[841]:= Series[Out[839], {x, Infinity, 9}] Out[841]= SeriesData[x, Infinity, 1 1 1 1 69311 {--, 0, -(---), 0, ----, 0, -(----), -(-----), 12 360 1260 1680 180 3892619 -------}, 1, 10, 1] 360 I.e., (with enough terms) we "recover" the divergent series from the convergent one! --rwg