Abel’s Theorem

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 $M$ .