I had read that if $g$ is an entire function then the radius of convergence of any power series expansion of $g$ is infinite, because we can enlarge the radius of the disc where the Taylor series represent the function as we want.
Also I know that if a function is real analytic then it can be uniquely extended to an holomorphic function. Thus if the analytic extension to the complex plane of a real-analytic function is entire we can conclude that the radius of convergence of any power series of $f$ is infinite. This is right?
Using the same reasoning we also can think that for any real-analytic function the radius of convergence of any power series expansion around a point only depends on the minimum distance to any singularity of the extension of $f$ to the complex plane. It is this reasoning correct?