Complex Power Series/Examples/n
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Example of Complex Power Series
The complex power series:
- $S = \ds \sum_{n \mathop \ge 0} n z^n$
has a radius of convergence of $1$.
Proof
Let $R$ denote the radius of convergence of $S$.
Thus:
\(\ds R\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {n - 1} n}\) | Radius of Convergence from Limit of Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \cmod {1 - \dfrac 1 n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | as $\sequence {\dfrac 1 n}$ is a basic null sequence |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $3 \ \text {(i)}$