Out[270]=Product[1 + E^(-((2*n*Pi)/5)) - (2*Cos[(2*n*Pi)/5])/E^((n*Pi)/5), {n,Infinity}] -> (Sqrt[5]*E^(Pi/60)*Gamma[1/4]^2)/(2*2^(3/4)*Pi^(3/2)) Amazing inaccurate numerics: In[272]:= Timing[N[%270, 105]] During evaluation of In[272]:= NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {185.8931420129281672288691448025888296859652775220603165483862095513741610526450692769280980058083317102024170890843}. NIntegrate obtained 0.00004366400531989029430161228066757194803765686030072049096264985465928621657418744962590402711855355493876269474449273`115. and 7.505823825434631924585246964567939510489203332210688578001209387505086611802644511529216233982994651935733171228298`115.*^-85 for the integral and error estimates. >> Out[272]= {29.6032, 1.65371039642291406870084030182905258859277665525891059324072211312882326107919099574691792549413518551613 -> 1.65371039642291406696569291506073467057126586745207656418583564992967124414571613904819032470934233477443} It's faster and hugely more accurate just to add the terms up: In[281]:= Timing[%270[[1]] /. Product[x_, L_] :> NProduct[x, {n, \[Infinity]}, WorkingPrecision -> 106, PrecisionGoal -> 105, NProductFactors -> 691]] During evaluation of In[281]:= NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in n near {n} = {861.8940068943142691044101719338395586499378157406267071093412114640734679433460889438059665960107225000717}. NIntegrate obtained 1.87009213033821895841004092891566274691306955453889921013030172021676370553701588750870721398048732949125`106.*^-189 and 2.602515810387861805818516463584710573116703810473833425997695963619028961690372962373958113185701592545913`106.*^-264 for the integral and error estimates. >> Out[281]= {26.7773, \ 1.653710396422914066965692915060734670571265867452076564185835649929671244145716139048190324709342334774} Upshot: N[%270/.Rule->Equal],n] gives False for nearly all n, discrediting the identity. --rwg