Existence of Radius of Convergence of Complex Power Series
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Theorem
Let $\xi \in \C$.
Let $\ds \map S 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 $\map S z$.
Absolute Convergence
Let $\map {B_R} \xi$ denote the open $R$-ball of $\xi$.
Let $z \in \map {B_R} \xi$.
Then $\map S z$ converges absolutely.
If $R = +\infty$, we define $\map {B_R} \xi = \C$.
Divergence
Let $\map { {B_R}^-} \xi$ denote the closed $R$-ball of $\xi$.
Let $z \notin \map { {B_R}^-} \xi$.
Then $\map S z$ is divergent.
Also see
- Existence of Interval of Convergence of Power Series for a proof of the same result in real numbers.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series