I could only get the odd terms: Out[368]= Sum[((1 - q)^2*q^k*(1 + q^(1 + 2*k)))/(1 - q^(1 + 2*k))^2, {k, 0, ∞}] = = ((1 + q)*QPochhammer[q^2, q^2]^3*QPochhammer[q^4, q^2])/ (QPochhammer[q, q^2]*QPochhammer[q^3, q^2]^3) Check q→1. Mathematica can't Limit the braindead QPochhammers. Punt them. In[369]:= depoch@% Out[369]= Sum[((1 - q)^2*q^k*(1 + q^(1 + 2*k)))/(1 - q^(1 + 2*k))^2, {k, 0, ∞}] = = ((1 + q)* Inactive[Product][1 - q^4*(q^2)^i, {i, 0, ∞}]* Inactive[Product][1 - (q^2)^(1 + i), {i, 0, ∞}]^3)/ (Inactive[Product][1 - q*(q^2)^i, {i, 0, ∞}]* Inactive[Product][1 - q^3*(q^2)^i, {i, 0, ∞}]^3) "Inactive" keeps them punted. In[370]:= Nest[prodcontract, %, 3] Out[370]= Sum[((1 - q)^2*q^k*(1 + q^(1 + 2*k)))/(1 - q^(1 + 2*k))^2, {k, 0, ∞}] = = (1 + q)*Inactive[Product][ ((1 - q^4*(q^2)^i)*(1 - (q^2)^(1 + i))^3)/((1 - q*(q^2)^i)*(1 - q^3*(q^2)^i)^3), {i, 0, ∞}] In[371]:= MapAt[Limit[#, q -> 1] &, %[[1]], 1] == MapAt[Limit[#, q -> 1] &, %[[2]], {2, 1}] Out[371]= Inactive[Sum][2/(1 + 2*k)^2, {k, 0, ∞}] == (1 + q)*Inactive[Product][(16*(1 + i)^3*(2 + i))/((1 + 2*i)*(3 + 2*i)^3), {i, 0, ∞}] In[372]:= Activate@% Out[372]= π^2/4 == (1 + q) π^2/8 (Left one q unsubstituted to avoid becoming True!) In[367]:= Series[%370 /. ∞ -> 8, {q, 0, 9}] Out[367] =1+2 q-q^2+3 q^4-6 q^5+3 q^6+8 q^7-16 q^8+7 q^9+O[q]^10= =1+2 q-q^2+3 q^4-6 q^5+3 q^6+8 q^7-16 q^8+8 q^9+O[q]^10 (Skimped on ∞ to avoid becoming True! But Macsyma could Taylor the infinite series.) Such a Lambertoid closed form Out[368,370] seems unfamiliar, both to me and Mathematica. I am too old and it is too young. —rwg Apologies for the "typesetting" imposed by the math-fun server.