Sum[(x^n)(∏_(k=1)^n)(((q-1))/((q^k)-1)), {n,0,∞}]

t1=With[{n=4}, (x^n)(∏_(k=1)^n)(((q-1))/((q^k)-1)) /. q->(E^ϵ)]

Series[t1,{ϵ,0,3}]

With[{n=4}, Normal@Series[(x^n)(∏_(k=1)^n)(((q-1))/((q^k)-1)) /. q->(E^ϵ), {ϵ,0,3}]]

Sum[Normal@Series[(x^n)(∏_(k=1)^n)(((q-1))/((q^k)-1)) /. q->(E^ϵ), {ϵ,0,0}], {n,0,∞}]

FullSimplify[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/x)/. F[ϵ_,x_] :> Sum[(x^n)(∏_(k=1)^n)(((q-1))/((q^k)-1)) /. q->(E^ϵ), {n,0,∞}]]

Simplify@Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/x), {ϵ,0,3}]

(x^n)(∏_(k=1)^n)Series[(((q-1))/((q^k)-1)) /. q->(E^ϵ),{ϵ,0,3}]

Simplify[(E^(-x-f[ϵ,x]))Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/x)/.F[ϵ_,x_] :> (E^(x+f[ϵ,x])), {ϵ,0,3}]]

Simplify[Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/(x F[ϵ,x]))/.F[ϵ_,x_] :> (E^(x+f[ϵ,x]))/.f[ϵ_,x_]->ϵ ϕ[ϵ,x], {ϵ,0,3}]]

Collect[Normal@Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/(x F[ϵ,x]))/.F[ϵ_,x_] :> (E^(x+f[ϵ,x]))/.f[ϵ_,x_]->((-(x^2))/4)ϵ+((x^3)/9)(ϵ^2)+(((-(x^4))/16)+((x^2)/48))(ϵ^3)+(((x^5)/25)-((x^3)/36))(ϵ^4)+(((-(x^6))/36)+((x^4)/32)-((x^2)/480))(ϵ^5)+(ϵ^6)ϕ[ϵ,x], {ϵ,0,7}],
ϵ, Expand]

Collect[Normal@Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/(x F[ϵ,x]))/.F[ϵ_,x_] :> (E^(x+f[ϵ,x]))/.f[ϵ_,x_]->((-(x^2))/4)ϵ+((x^3)/9)(ϵ^2)+(((-(x^4))/16)+((x^2)/48))(ϵ^3)+(((x^5)/25)-((x^3)/36))(ϵ^4)+(((-(x^6))/36)+((x^4)/32)-((x^2)/480))(ϵ^5)+(ϵ^6)ϕ[ϵ,x], {ϵ,0,7}],
ϵ, Expand]

-Integrate[-((7 (x^2))/360)+((x^4)/6)-((x^6)/7),x]

Collect[Normal@Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/(x F[ϵ,x]))/.F[ϵ_,x_] :> (E^(x+f[ϵ,x]))/.f[ϵ_,x_]->((-(x^2))/4)ϵ+((x^3)/9)(ϵ^2)+(((-(x^4))/16)+((x^2)/48))(ϵ^3)+(((x^5)/25)-((x^3)/36))(ϵ^4)+(((-(x^6))/36)+((x^4)/32)-((x^2)/480))(ϵ^5)+(((7 (x^3))/1080)-((x^5)/30)+((x^7)/49))(ϵ^6)+(ϵ^7)ϕ[ϵ,x], {ϵ,0,8}],
ϵ, Expand]

-Integrate[-((17 x)/40320)+((17 (x^3))/320)-((5 (x^5))/24)+((x^7)/8),x]

Collect[Normal@Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/(x F[ϵ,x]))/.F[ϵ_,x_] :> (E^(x+f[ϵ,x]))/.f[ϵ_,x_]->((-(x^2))/4)ϵ+((x^3)/9)(ϵ^2)+(((-(x^4))/16)+((x^2)/48))(ϵ^3)+(((x^5)/25)-((x^3)/36))(ϵ^4)+(((-(x^6))/36)+((x^4)/32)-((x^2)/480))(ϵ^5)+(((7 (x^3))/1080)-((x^5)/30)+((x^7)/49))(ϵ^6)+(((17 (x^2))/80640)-((17 (x^4))/1280)+((5 (x^6))/144)-((x^8)/64))(ϵ^7)+(ϵ^8)ϕ[ϵ,x], {ϵ,0,9}],
ϵ, Expand]

-Integrate[((809 (x^2))/181440)-((9 (x^4))/80)+((x^6)/4)-((x^8)/9),x]

Collect[Normal@Series[((F[ϵ,(E^ϵ)x]-(((E^ϵ)-1)x+1)F[ϵ,x])/(x F[ϵ,x]))/.F[ϵ_,x_] :> (E^(x+f[ϵ,x]))/.f[ϵ_,x_]->((-(x^2))/4)ϵ+((x^3)/9)(ϵ^2)+(((-(x^4))/16)+((x^2)/48))(ϵ^3)+(((x^5)/25)-((x^3)/36))(ϵ^4)+(((-(x^6))/36)+((x^4)/32)-((x^2)/480))(ϵ^5)+(((7 (x^3))/1080)-((x^5)/30)+((x^7)/49))(ϵ^6)+(((17 (x^2))/80640)-((17 (x^4))/1280)+((5 (x^6))/144)-((x^8)/64))(ϵ^7)+(-((809 (x^3))/544320)+((9 (x^5))/400)-((x^7)/28)+((x^9)/81))(ϵ^8)+(ϵ^9)ϕ[ϵ,x], {ϵ,0,10}],
ϵ, Expand]

-Integrate[((31 x)/725760)-((527 (x^3))/24192)+((59 (x^5))/288)-((7 (x^7))/24)+((x^9)/10),x]

Factor[-((31 (x^2))/1451520)+((527 (x^4))/96768)-((59 (x^6))/1728)+((7 (x^8))/192)-((x^10)/100)]

Comparing with Quesne’s arXiv:math-ph/0305003:

(Subscript[c, k_])[q_] := ((((1-q))^k)/(k(1-(q^k))));
With[{n=5}, Collect[
(E^(-x))Normal@Series[Exp[Sum[(Subscript[c, k])[(E^ϵ)](x^k),{k, 1,n+1}]], {ϵ,0,n}],
ϵ,Expand
]]

(Subscript[c, k_])[q_] := ((((1-q))^k)/(k(1-(q^k))));
With[{n=9}, Collect[
Normal@Series[Sum[(Subscript[c, k])[(E^ϵ)](x^k),{k, 1,n+1}], {ϵ,0,n}],
ϵ,Expand
]]