here's a MAPLE script... E := exp(1); g := (z) -> E * (GAMMA(z+1/2)/sqrt(2*Pi))^(1/z); series( g(z), z = infinity, 50 ); #Now to overcome the fact MAPLE is a total idiot: subs(exp(-1/2)*exp(1/2)=1, %); subs(1/exp(-1/2)/exp(1/2)=1, %); subs(ln(1/exp(-1/2)/exp(1/2))=0, %); subs(ln(1)=0, %); subs(exp(-1)*exp(1)=1, %); subs(exp(1)*exp(-1)=1, %); subs(z = 1/R, %); simplify(%); #to get at last this SERIES... series( %, R, 48 ); 1/R -1/24*R +19/5760*R^3 -2561/2903040*R^5 +874831/1393459200*R^7 -319094777/367873228800*R^9 +47095708213409/24103053950976000*R^11 -751163826506551/115694658964684800*R^13 +... The first interesting amazing observation is, f only involves ODD powers of z (or of R=1/z). We can now convert f to a CONTINUED FRACTION. I find... g(z)/z = 1 - y/(24 + y/(10/19 + y/(181944/18029 + y/(59099062/376343621 + ... )))) where y = z^(-2) = R^2 The coefficients in floating point instead of exact rational form are coeff#......value ======......===== 1 ....... 24.0 2 ....... 0.5263157895 3 ....... 10.09174108 4 ....... 0.1570348445 5 ....... 7.183113315 6 ....... 0.08255654793 7 ....... 5.815365631 8 ....... 0.05303121627 9 ....... 4.991323711 10 ....... 0.03780359727 11 ....... 4.429552829 12 ....... 0.02873111220 13 ....... 4.016881221 14 ....... 0.02280569189 15 ....... 3.698138694 16 ....... 0.01868054686 17 ....... 3.442901964 18 ....... 0.01567044292 19 ....... 3.232881310 20 ....... 0.01339321448 21 ....... 3.056353642 22 ....... 0.01162041072 23 ....... 2.905423085 24 ....... 0.01020783673 25 ....... 2.774557162 ... The second interesting amazing observation is, all these coefficients are POSITIVE. The odd numbered and even numbered coefficients evidently behave differently. It looks vaguely as though they might be tending toward two positive constants in which case you could cut off the continued fraction and replace the rest with a periodic CF, whose limit L is known in closed form by solving a quadratic equation y/(a+y/(b+L))=L where in our case, the positive root clearly must be chosen. This also would prove convergence for all positive z by "Van Vleck's CF convergence theorem." Or they might be gradually shrinking toward 0. If they all are positive and shrink slowly enough toward 0 that their sum diverges, then again this would prove convergence of the CF for each positive z by "Van Vleck's CF convergence theorem" mentioned here http://en.wikipedia.org/wiki/Convergence_problem It looks to me like this happens (although that mere appearance proves nothing).