Radius of Convergence

The radius of convergence of a power series centered at 0 with complex coefficients is the number such that the power series converges absolutely for and diverges for . For real , the interior of the interval of convergence is when . When , either endpoint may also belong to the interval of convergence. For example, the geometric series converges precisely for .

For complex numbers , the condition defines an open disk of radius when . The Cauchy-Hadamard theorem gives

(1)

(2)

where denotes the supremum limit, , and .

To prove absolute convergence, fix with and choose a finite with , also when . Since , for all sufficiently large ,

(3)

The right side is the term of a convergent series because , so the comparison test proves absolute convergence. At the power series equals . If , the root test gives divergence, since the supremum limit of the th roots of the absolute values of the terms is .