Abel's Theorem - Theorem

Theorem

Let a = {a_k: k ≥ 0} be any sequence of real or complex numbers and let

be the power series with coefficients a. Suppose that the series converges. Then

where the variable z is supposed to be real, or, more generally, to lie within any Stoltz angle, that is, a region of the open unit disk where

for some M. Without this restriction, the limit may fail to exist.

Note that is continuous on the real closed interval for t < 1, by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that is continuous on .