Power Series

A power series in a variable is an infinite sum of the form

where are integers, real numbers, complex numbers, or any other quantities of a given type.

Pólya conjectured that if a function has a power series with integer coefficients and radius of convergence 1, then either the function is rational or the unit circle is a natural boundary (Pólya 1990, pp. 43 and 46). This conjecture was stated by G. Polya in 1916 and proved to be correct by Carlson (1921) in a result that is now regarded as a classic of early 20th century complex analysis.

For any power series, one of the following is true:

1. The series converges only for .

2. The series converges absolutely for all .

3. The series converges absolutely for all in some finite open interval and diverges if or . At the points and , the series may converge absolutely, converge conditionally, or diverge.

To determine the interval of convergence, apply the ratio test for absolute convergence and solve for . A power series may be differentiated or integrated within the interval of convergence. Convergent power series may be multiplied and divided (if there is no division by zero).

converges if and diverges if .