Power Series — Radius and Interval of Convergence

GRE Math Subject · Axiom Academy

LESSON Power Series -- Radius and Interval of Convergence Where power series converge, common Taylor/Maclaurin series, and term-by-term operations General power series centered at a: The constants are the coefficients . The series is a function of x that may converge on some interval around x = a. Three possible outcomes for convergence: R = 0: Converges only at x = a. 0 < R < : Converges on an interval (a - R, a + R), with endpoints needing separate checks. R = : Converges for all x (entire real line). Finding the Radius of Convergence R The series converges absolutely when this limit is < 1, i.e., when . So (or R = if L = 0, or R = 0 if L = ). Then R = 1/L (same conventions for L = 0 or ). Hadamard's formula (most general): This always works, even when the ratio/root limits don't exist. Finding the Interval of Convergence Use the ratio or root test to find R. The series converges on the open interval (a - R, a + R). Check the left endpoint x = a - R by substituting and testing the resulting numerical series. Check the right endpoint x = a + R by substituting and testing. Combine: the interval is one of (a-R, a+R), [a-R, a+R), (a-R, a+R], or [a-R, a+R]. GRE trap: Never forget to check endpoints. The ratio/root test is always inconclusive at |x - a| = R. You must test each endpoint individually using convergence tests for numerical series (alternating series test, p-series, comparison, etc.). Essential Taylor/Maclaurin Series (Memorize These)

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