Rules for determining interval of convergence of power series of rational function

This article describes a rule, set of rules, or procedure, for determining the set of values for a parameter where a series or integral defined in terms of that parameter converges.
See other rules for determining interval of convergence

Goal

Suppose we have a rational function:

$f(x) := \frac{p(x)}{q(x)}$

Assume that $p$ and $q$ have no common divisors.

Consider a point $x_0$ such that $q(x_0) \ne 0$. Then, $f$ is locally analytic at $x_0$: the Taylor series of $f$ at $x_0$ converges to $x_0$ in an open interval centered at $x_0$. In fact, this open interval coincides with the interval of convergence of the Taylor series, i.e., the Taylor series converges to $f$ wherever it does converge.

The interval of convergence is an open interval. Below, we give the rules to determine this interval.

Case that $q$ is a constant, so $f$ is polynomial

In this case, the interval of convergence is all of $\R$.

Case that $q$ splits completely over $\R$

In case $q$ is expressible as a product of (possibly repeated) linear factors, the radius of convergence of the power series for $f$ is given as the minimum of the distances between $x_0$ and the zeros of $q$. Note that this minimum is positive, as $q$ has finitely many zeros, and none of them equals $x_0$ by assumption $q(x_0) \ne 0$.

The interval of convergence is the open interval centered at $x_0$ with the above radius of convergence.

Case that $q$ has irreducible quadratic factors

Using complex number language: split $q$ completely as a product of (possibly repeated) linear factors over the complex numbers. Then, the radius of convergence of the power series for $f$ is given as the minimum of the distances between $x_0$ and the (real and non-real) zeros of $q$. Here, the distance between two complex numbers is defined as the modulus of their difference.

Using real number language: Take the minimum of the following: for each linear factor, the distance from $x_0$ to the corresponding zero, and for an irreducible quadratic factor $ax^2 + bx + c$, the value $\sqrt{x_0^2 + bx_0/a + c/a}$.

The interval of convergence is the open interval centered at $x_0$ with the above radius of convergence.