Mathematica should be able to get Product[(3^(-(1/2) + 3*n)*Gamma[2/21 + n]*Gamma[4/21 + n]*Gamma[5/7 + n])/ (2*Pi*Gamma[3*n]), {n, Infinity}] == (BarnesG[4/3]*BarnesG[5/3])/ (BarnesG[23/21]*BarnesG[25/21]*BarnesG[12/7]) In[885]:= N[Rule @@ %] During evaluation of In[885]:= NIntegrate::slwcon:Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >> During evaluation of In[885]:= NIntegrate::ncvb:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {7273.96592}. NIntegrate obtained -0.000408638628 and 8.024253968625528`*^-8 for the integral and error estimates. >> Out[885]= 0.9878943769692339 -> 0.9878939404389084 simply by expressing the finite product using BarnesG, then taking the limit using "Barnes' Stirling's formula". --rwg