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) = sb(1)(x)t + b(2)(x)t(2) + ... of analytic curves in C x C-n passing through the origin, Conv(phi)(f) of a formal power series f (y, t, x) is an element of C[[y, t, x]] is defined to be the set of all s is an element of 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 subset of C there exists a divergent formal power series f(y, t, x) is an element of C[[y, t, x]] such that E = Conv(phi)(f) if and only if E is an F-sigma set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case phi(s)(t, x) = st.