  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 holomorphic
functions on a domain $\Omega$ in $\mathbb{C}$. A subset $E$ of $\Omega$ is
said to be a convergence set in $\Omega$ if there is a series $f(z,t)$ such
that $E$ is exactly the set of points $z$ for which $f(z,t)$ converges as a
power 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 of
polynomially convex compact subsets. We prove that a subset of $\mathbb{C}$ is
a convergence set if and only if it is $\sigma$-convex.
