Existence of Radius of Convergence of Complex Power Series

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Theorem

Let $\xi \in \C$.

Let $\displaystyle S \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n $ be a (complex) power series about $\xi$.

Then there exists a radius of convergence $R \in \overline \R$ of $S \paren z$.


Absolute Convergence

Let $B_R \paren \xi$ denote the open $R$-ball of $\xi$.

Let $z \in B_R \paren \xi$.

Then $S \paren z$ converges absolutely.


If $R = +\infty$, we define $B_R \paren \xi = \C$.


Divergence

Let ${B_R}^- \paren \xi$ denote the closed $R$-ball of $\xi$.

Let $z \notin {B_R}^- \paren \xi$.

Then $S \paren z$ is divergent.


Also see


Sources