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A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y={\phi }_s(t,x)=sb₁\left(x\right)t+b₂\left(x\right)t²+\cdots$ of analytic curves in ℂ × ℂⁿ passing through the origin, $Con{v}_{\phi }\left(f\right)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f({\phi }_s(t,x),t,x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E=Con{v}_{\phi }\left(f\right)$ if and only if...