Complex Power Series/Examples/n

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 $3 \ \text {(i)}$