Power Series in Two Variables and Radius of Convergence

Question

Let $\alpha > 0$, $\beta > 0$, and assume that the power series with real coefficients \begin{equation} \sum_{n,m = 0}^{\infty} a_{n,m} x^{n} y^{m} \end{equation} is absolutely convergent for every real $x, y$ such that $|x| < \alpha$, $|y| < \beta$. Then you can rearrange the terms of the series in ascending powers of $x$ as \begin{equation} \sum_{n = 0}^{\infty} f_n(y) x^{n}, \end{equation} where the function $f_n:(-\beta,\beta) \rightarrow \mathbb{R}$ are analytic functions. For every fixed $y \in (-\beta,\beta)$, let $R(y)$ be tha radius of convergence of the series in $x$ \begin{equation} \sum_{n = 0}^{\infty} f_n(y) x^{n}. \end{equation} Is $R$ a continuous function of $y$? I guess that, generally speaking, the answer is negative, but I have no counterexample.

Answer 1

Consider $\sum_{n=1}^{\infty}(2^n\sin y)x^n.$ If $y\in \mathbb R \setminus \pi\mathbb Z,$ then

$$\limsup_{n\to \infty}|2^n\sin y|^{1/n} = 2.$$

Hence for those values of $y,$ the radius of convergence is $1/2.$ On the other hand, if $y \in \pi\mathbb Z,$ then the series vanishes identically and the radius of convergence is $\infty.$