Complex Power Series/Examples/n

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Example of Complex Power Series

The complex power series:

$S = \displaystyle \sum_{n \mathop \ge 0} n z^n$

has a radius of convergence of $1$.


Proof

Let $R$ denote the radius of convergence of $S$.

Thus:

\(\displaystyle R\) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {n - 1} n}\) Radius of Convergence from Limit of Sequence
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \cmod {1 - \dfrac 1 n}\)
\(\displaystyle \) \(=\) \(\displaystyle 1\) as $\sequence {\dfrac 1 n}$ is a basic null sequence

$\blacksquare$


Sources