Cauchy–Hadamard theorem

The Cauchy–Hadamard theorem states that a power series has radius of convergence equal to the reciprocal of the limit superior of the nth roots of its coefficient magnitudes, with the usual conventions for zero and infinity.

Radius of convergence: The radius of the largest disk centered at a power series’ center where the series converges. The theorem computes this radius directly from the coefficients.
Absolute convergence: Convergence of a series whose terms’ absolute values also form a convergent series. Inside the stated radius, power series converge absolutely.
Augustin-Louis Cauchy: A French mathematician whose work helped establish modern analysis and complex function theory. His name is attached to the theorem and to foundational convergence methods.
Ratio test: A convergence test based on the limit of ratios of successive term magnitudes. It can give the same radius when coefficient ratios behave regularly, but may fail when they oscillate.
Limit superior: The limit of the supremum of each sequence tail, allowing oscillating sequences to have a precise asymptotic bound. It captures the coefficients’ upper asymptotic growth even when nth roots do not converge.
Complex plane: The plane of complex numbers, represented by real and imaginary coordinates. A complex power series converges on a disk measured by distance from its center.
Jacques Hadamard: A French mathematician known for contributions to analysis, geometry, and mathematical physics. His name is attached to the theorem’s coefficient-growth formula.
Abel's theorem: A theorem relating boundary limits of a power series to the limit of its sum from within its disk of convergence. It addresses some boundary behavior that the radius formula alone cannot settle.
Root test: A convergence test that uses the limit superior of nth roots of term magnitudes. Applied to power-series terms, it yields the theorem’s radius formula.
Sequence: An ordered list of terms indexed by integers or another ordered set. The coefficient magnitudes’ nth roots form the sequence whose limit superior appears.

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Cauchy condensation testRelated: The condensation test can establish convergence of coefficient series with logarithmic decay.