Finding Radius of Convergence

Complex Analysis · Axiom Academy

EXAMPLE Finding the Radius of Convergence Determine where a complex power series converges using the Cauchy–Hadamard formula and the nearest singularity. Find the radius of convergence R of the complex power series In other words: for which complex numbers z does this series converge? Nicely done. You found the radius of convergence two independent ways and got the same answer. Cauchy–Hadamard formula: . This is the most general way to find the radius — it always applies. Ratio-test shortcut: when the limit exists, . Here that gives , agreeing with the root test. Nearest-singularity rule: R equals the distance from the center to the nearest singularity of the function the series represents. Since e^z is entire, there is no singularity, so . Disk of convergence: in general the series converges absolutely for and diverges for ; behavior on the circle |z| = R must be checked separately. Here the "disk" is the whole plane. The big idea: the radius of convergence is dictated by the nearest singularity of the function being represented — a direct bridge between a series and the analytic function behind it.

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