Power Series and Convergence Radius

Real Analysis · Axiom Academy

LESSON Power Series and Convergence Radius Every power series lives on an interval centered at a , with a radius R that says exactly where it converges — and endpoints that play by their own rules. Fix a center a . For the power series there is a number , the radius of convergence , that splits the real line into three zones by the distance |x-a| from the center. at |x-a| = R (the two endpoints), convergence must be tested separately. 2. Computing R : Cauchy–Hadamard The radius is read straight off the coefficients. The root test always works; the ratio test is often easier when the limit exists. Both measure how fast |c_n| grows. Cauchy–Hadamard (root test) — always valid Ratio test — when this limit exists Watch the running values for , where c_n = 1/n . Both the ratio and the root |c_n|^ 1/n home in on 1 , so R = 1 . Ratio: . Root: , so 1/R = 1 . Either way R = 1 . 3. The Endpoints Decide for Themselves Inside the radius we have absolute convergence; outside, divergence. But at |x-a| = R the test is silent — the two endpoints can converge, diverge, or split , one each. The interval of convergence is whatever genuine interval results. Take with R = 1 , centered at a = 0 . Its endpoints are x = -1 and x = 1 , and they behave oppositely — watch the partial sums. is the alternating harmonic series. By the alternating series test it converges (to ). is the harmonic series. Its partial sums grow without bound, so it diverges. 4. Coefficient Growth Sets the Radius

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