radiusofconvergence

Radius of convergence is the nonnegative number R associated with a power series Σ a_n (z − z0)^n. It describes the largest distance from the center z0 within which the series converges to a finite sum. In the complex plane, the series converges absolutely for all z with |z − z0| < R and diverges for |z − z0| > R. For a real-variable series in x about x0, the interval of convergence is (x0 − R, x0 + R) with the same interpretation.

The radius is determined by the Cauchy–Hadamard formula: R = 1 / limsup_{n→∞} (|a_n|)^{1/n}, where the limsup may be 0, in which case R = ∞ and the series converges for all z. If the limit L = lim_{n→∞} |a_{n+1}|/|a_n| exists, then R = 1/L. Equivalently, R = lim_{n→∞} |a_n|/|a_{n+1}| when that limit exists.

Endpoints are not determined by R alone; convergence at points with |z − z0| = R may occur for some boundary points and fail for others, so those points require separate testing.

Within the disk |z − z0| < R, the series defines a holomorphic (analytic) function, and R marks the boundary of its domain of analyticity. The concept extends similarly to several variables, though the geometry becomes more complex.

Examples: Σ z^n centered at 0 has R = 1. The real series Σ x^n also has R = 1 about x0 = 0. The series Σ n! z^n has R = 0, converging only at z = 0. The series Σ z^n/n has R = 1.