Testing Series Convergence

Complex Analysis · Axiom Academy

EXAMPLE Testing Series Convergence Use the ratio test to find the radius of convergence of a complex power series — then read its boundary. For which complex numbers z does the power series converge? Find its radius of convergence, then decide what happens on the boundary circle. The series converges on the open disk |z| < 1 and diverges outside it, so the radius is 1. On the boundary circle it converges everywhere except the single point z = 1. Nice work. One ratio-test limit pinned the radius, and a single by-hand check settled the boundary. Match the test to the form: a power series with a clean a_n calls for the ratio (or root) test on |a_ n+1 /a_n| — both give the same limit, so use whichever is easier. The limit names the radius: here , so L < 1 becomes |z| < 1 and the radius of convergence is R = 1 . The boundary is a separate question: when L = 1 the ratio test is inconclusive, so the circle |z| = 1 must be checked point by point. At z = 1 the series is the harmonic series , which diverges ; at z = -1 it is the alternating harmonic series, which converges . Region of convergence: converges for |z| < 1 , diverges for |z| > 1 , and on |z| = 1 converges everywhere except z = 1 . (Inside the disk it sums to .) Choose a test, take one limit, read the radius — then treat the boundary circle as its own little problem.

This is the written version of the interactive lesson above. See the full Complex Analysis course.