Abel's theorem

A power series defines a function safely inside its interval of convergence. But what about the very edge — the endpoint, where convergence is delicate? If the series happens to converge there too, does the function it sums to match the limit you approach from inside? Abel's theorem says yes: the function joins smoothly to its endpoint value, with no jump.

Statement: suppose sum c_n converges (to S). Then the power series f(x) = sum c_n x^n, which has radius of convergence at least 1, satisfies lim_{x->1^-} f(x) = S. In words, if the series converges at the boundary point x = 1, the sum equals the limit of f as x approaches 1 from the left inside the interval. The same holds at either endpoint after rescaling.

The value of the theorem is that it lets you evaluate the sum of a series by computing a limit of a known function. The converse is false and important: f(x) can have a limit as x -> 1^- even when sum c_n diverges (for instance sum (-1)^n has f(x) = 1/(1+x) -> 1/2, yet the series does not converge). Recovering convergence of the series from existence of the limit needs extra hypotheses — those are Tauberian theorems.