Radius of convergence

The radius of convergence of a power series is the distance from its center to the boundary of the largest disk in which the series converges. Inside that disk it converges absolutely; outside it diverges.

Ratio test: A convergence test that examines the limit of ratios of successive terms in a series. Applied to power-series terms, it often yields the radius through the coefficients’ growth.
Power series: An infinite series of powers of a variable’s difference from a fixed center, multiplied by coefficients. The radius of convergence is defined for this class of series.
Analytic function: A function locally represented by a convergent power series at every point in its domain. A Taylor series represents its function at least until its center’s nearest complex singularity.
Formal power series: A sequence of coefficients treated algebraically as an infinite power series, without requiring numerical convergence. A formal series has no convergence radius unless analytic convergence is separately considered.
Root test: A convergence test based on the limit superior of the nth roots of a series’ term magnitudes. It gives a radius formula directly from the coefficients, including when ratios fail.
Taylor series: A power series whose coefficients are determined by the derivatives of a function at a center. Its radius sets the disk around the expansion center where the Taylor series converges.
Taylor's theorem: A theorem relating a function to a finite Taylor polynomial and a remainder term. The convergence radius helps determine where an infinite Taylor expansion can represent a function.
Interval of convergence: The real interval on which a real power series converges. It is the real-variable counterpart; endpoint behavior can make it differ from the open interval set by the radius.
Cauchy–Hadamard theorem: A theorem that characterizes a power series’ radius of convergence using the limit superior of nth-root coefficient magnitudes. It states the general coefficient formula for the radius, including infinite and zero cases.
Complex plane: The plane of complex numbers, with real and imaginary parts as perpendicular coordinates. For complex power series, convergence regions are disks measured with complex distance.

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