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We consider convergence sets of formal power series of the form$f(z,t)=\sum_{n=0}^{\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphicfunctions on a domain $\Omega$ in $\mathbb{C}$. A subset $E$ of $\Omega$ issaid to be a convergence set in $\Omega$ if there is a series $f(z,t)$ suchthat $E$ is exactly the set of points $z$ for which $f(z,t)$ converges as apower series in a single variable $t$ in some neighborhood of the origin. A$\sigma$-convex set is defined to be the union of a countable collection ofpolynomially convex compact subsets. We prove that a subset of $\mathbb{C}$ isa convergence set if and only if it is $\sigma$-convex.

On Convergence Sets of Power Series with Holomorphic Coefficients