CONVERGENCE DOMAINS OF POWER SERIES WITH RATIONAL COEFFICIENTS

Abstract

The Taylor’s expansion enables us to expand an infinitely differentiable function into a power series. The opposite problem is the summation of a power series. Before calculating the sum of a power series, we need to find its domain of convergence because only on that domain does the sum of the series exist. This leads to the problem of finding the radius of convergence of the power series.

We know that if  when  tends to infinity, two power series with coefficients  will have the same radius of convergence. This allows us to identify which types of power series have the same radius of convergence by comparing their coefficients as  tends to infinity. In [5], the authors chose the power function  as an intermediary in comparing the extremely small quantities when  tends to result in zero. In this article, we choose the coefficient  as a standard to determine the types of power series that have the same radius of convergence with the series with factor . Then we go on to indicate that in this class, the power series with rational coefficients have the same domain of convergence with the power series with factor .