Abel’s Theorem

Posted on May 25, 2016 by

Abel’s Theorem is a useful theorem in analysis.

Let (a_k) be any sequence in \mathbb{R} or \mathbb{C}. Let G_a(z)=\sum_{k=0}^\infty a_k z^k. Suppose that the series \sum_{k=0}^\infty a_k converges. Then
\lim_{z\to 1^-}G_a(z)=\sum_{k=0}^\infty a_k, where z is real, or more generally, lies within any Stolz angle, i.e.\ a region of the open unit disk where |1-z|\leq M(1-|z|) for some M.

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