Circle of Convergence

circle of convergence [′sər·kəl əv kən′vər·jəns]

(mathematics)

The region in which a power series possesses a limit.

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Circle of Convergence 

The circle of convergence of a power series

(*) a₀ + a₁(z - z₀)² + a₂(z - z₀)² + …

is a circle ǀz – z₀ǀ = R in the plane of the complex variable z such that the series (*) con verges if ǀz – aǀ < R and diverges if ǀz – aǀ > R (the series may both converge and diverge at points on the circle ǀz – z₀ǀ = R). Every power series converges in the entire plane (for any z), has a circle of convergence of finite radius R, or converges only when z = Z₀. The series (*) converges to an analytic function in the circle of convergence. The number R is called the radius of convergence of the series (*) and determined from the Cauchy-Hadamard formula

If z₀ = x₀ is a real number, then the part of the real axis Ox lying in the circle of convergence is called the interval of convergence.