Abel's theorem (power series)

Abel's theorem states that if a power series converges at 1, its sum for real inputs approaching 1 from below tends to the series' sum at 1.

Abel transformation: A summation-by-parts identity that rewrites a weighted finite sum using partial sums and successive weight differences. It converts the power-series tail into terms controlled by bounded partial sums.
Power series: An infinite series of powers of a variable, with fixed coefficients. The theorem concerns the boundary behavior of this class of series.
Alternating harmonic series: The convergent series whose terms alternate in sign and have magnitudes equal to reciprocals of positive integers. Its power series gives a classic limit argument for evaluating the endpoint sum.
Abel's theorem (complex analysis): A result in complex analysis connecting boundary values of analytic functions with convergent power-series coefficients. This related theorem is often stated in broader complex-variable settings.
Uniform convergence: A sequence of functions converges uniformly when one error bound works across its entire domain. Abel's theorem reaches the boundary without assuming uniform convergence on the full interval.
Radius of convergence: The radius of the largest disk centered at a power series' center where the series converges. Convergence at the boundary point 1 is the extra hypothesis beyond interior convergence.
Taylor series: A power series whose coefficients are determined by derivatives of a function at a center. Abel's theorem can relate a Taylor series' boundary sum to interior function values.
Tauber's theorem: A theorem that derives ordinary convergence from an Abelian limit under additional conditions. It reverses Abelian reasoning by recovering series convergence from a damped limit.
Cauchy criterion: A criterion stating that a sequence converges exactly when its terms eventually become arbitrarily close to one another. Cauchy tails of the series bound the boundary error in the proof.
Absolute convergence: A series converges absolutely when the series of absolute values of its terms converges. Absolute convergence at 1 is sufficient, but Abel's theorem also applies to conditional convergence.

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