Abel’s theorem for power series relates a limit of a power series to the sum of its coefficients.
0.1 Abel’s Theorem
Let the Taylor series \(G(x)=\sum_{k=0}^\infty a_{k}x^k\) be a power series with real coefficients \(a_k\) with radius of convergence 1. Suppose that the series \(\sum_{k=0}^\infty a_{k}\) converges. Then \(G(x)\) is continuous frm the left at \(x=1\), that is \[ \lim_{x\rightarrow 1^- }G(x)=\sum_{k=0}^\infty a_{k.} \]
The same theorem holds for complex power series with \(x\in\mathbb{C}\) provided that \(x\rightarrow 1\) entirely within a single Stolz section, that is, a region of the open unit disk where \[ |1-z|\leq M(1-|z|), \] for some fixed finite \(M>1\).