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

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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.