Mathematica doesn't expand the series of this function (I'm using Wolfram Cloud version)
\[Phi][z_]:=Sum[(-1)^(j+1)z^j/(1-z^j),{j,1,Infinity}];
Series[\[Phi][z]/(1-\[Phi][z]),{z,0,10}]
Is there any other method to do it?
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1 Answer
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One solution is to drop all terms in the definition of ϕ that do not affect the series expansion up to the requested order:
ϕ[z_,n_]:=Sum[(-1)^(j+1)z^j/(1-z^j),{j,1,n}];
With[{n=10}, Series[ϕ[z,n]/(1-ϕ[z,n]),{z,0,n}]]
(* z+z^2+3 z^3+4 z^4+7 z^5+14 z^6+23 z^7+39 z^8+71 z^9+124 z^10+O[z]^11 *)
See also A003242.
