Abel's theorem
/ AH-bell /
A power series tells you nothing for certain about its own boundary circle — convergence inside the disk says nothing about the rim. But suppose you get lucky and the series happens to converge at a particular boundary point. Does its value there agree with the limit of the function as you approach from inside? Abel's theorem says yes, provided you approach along a sensible path. It is the bridge that lets you read a boundary value off the series whenever the series cooperates at the boundary.
Precisely, suppose the power series f(z) = sum a_n z^n has radius of convergence 1 and, at a boundary point — take z = 1 for concreteness — the numerical series sum a_n actually converges to a value S. Then f(z) tends to S as z approaches 1 from inside the disk, as long as z stays within a 'Stolz angle' (a sector that meets the circle non-tangentially, not sliding in along the boundary). In short: f(1) in the sense of the convergent series equals the radial (or non-tangential) limit of f. The series value and the function's boundary limit match. The non-tangential restriction is genuinely needed — approaching tangentially can spoil the conclusion.
Abel's theorem is what makes certain beautiful evaluations rigorous. The series for log: sum (-1)^(n+1) z^n / n equals -Log(1 - z) inside the disk, and at z = 1 the series becomes the alternating harmonic series 1 - 1/2 + 1/3 - ... which converges; Abel's theorem certifies that its sum is the boundary limit, namely log 2. Likewise the Leibniz series 1 - 1/3 + 1/5 - ... = pi/4 is justified by applying Abel's theorem to the arctangent series at z = 1. Its converse is FALSE in general — a radial limit can exist while the series at the boundary diverges (an Abel-summable but non-convergent series), which is the starting point of summability theory and Tauberian theorems.
Take f(z) = sum_{n>=1} (-1)^(n+1) z^n / n = -Log(1 - z) on |z| < 1. At z = 1 the series is 1 - 1/2 + 1/3 - 1/4 + ..., which converges. Abel's theorem says its sum equals lim_{z -> 1} f(z) = -Log(0... ) computed as the boundary limit = log 2. So the alternating harmonic series sums to log 2, made rigorous by Abel.
Abel's theorem licenses passing the limit z -> 1 inside the series, turning a boundary series into a clean closed-form sum.
The converse fails: a function can have a radial boundary limit while the corresponding series diverges there (it is then only 'Abel summable'). Abel's theorem goes one way only — convergence of the series implies the limit, not the reverse.