Consider a random power series Σ0∞ cn zn, that is, with coefficients {cn}0∞ chosen independently at random from the complex plane. What is the radius of convergence of such a series likely to be?

One approach to this question is to let the {cn}0∞ be independent random variables on some probability space. It turns out that, with probability one, the radius of convergence is constant. Moreover, if the cn are symmetric and have the same distribution, then the circle of convergence is almost surely a natural boundary for the analytic function given by the power series (See [1, Ch. IV, Section 3]). Our treatment of the question will be elementary and will not use these facts.