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Radius & Interval of Convergence

Determining the radius and interval of convergence of a power series.

Mapped to AP College Board # LIM-8, LIM-8.D, LIM-8.D.1, LIM-8.D.2, LIM-8.D.3, LIM-8.D.4, LIM-8.D.5, LIM-8.D.6

Power series allow us to represent associated functions on an appropriate interval. Determine the radius of convergence and interval of convergence for a power series. A power series is a series of the form ∑an (x-r) where n has values from 0 to ∞, where n is a non-negative integer, {an} is a sequence of real numbers, and r is a real number. If a power series converges, it either converges at a single point or has an interval of convergence. The ratio test can be used to determine the radius of convergence of a power series. The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence. If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval. The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series.