Domain of Convergence

the set of values of a variable x for which a series of functions

converges. The domain of convergence has a very simple form for power series. If a power series is considered for real values of the independent variable, then its domain of convergence is a single point, an interval (seeINTERVAL OF CONVERGENCE OF A POWER SERIES), which may contain one or both end points, or the entires-axis. If, however, complex values of the independent variable are also considered, then the domain of convergence of a power series is a single point, the interior of some circle (the circle of convergence), the interior of a circle and some points on the circumference, or the entire complex plane. Other types of series may have more complicated domains of convergence. For example, for a series of Legendre polynomials in the complex domain, the domain of convergence is the interior of an ellipse with foci at points — 1 and +1.

The domain of convergence is also defined for other processes. For example, the domain of convergence of an improper integral dependent on a parameter is understood to be the set of values of the parameter for which the given improper integral converges.