Complex Power Series/Examples/3^n-1 over 2^n+1
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Example of Complex Power Series
Let $\sequence {a_n}$ be the sequence defined as:
- $a_n = \dfrac {3^n - 1} {2^n + 1}$
The complex power series:
- $S = \ds \sum_{n \mathop \ge 0} a_n z^n$
has a radius of convergence of $\dfrac 2 3$.
Proof
Let $R$ denote the radius of convergence of $S$.
By Radius of Convergence from Limit of Sequence:
- $R = \ds \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }$
Thus:
\(\ds a_n\) | \(=\) | \(\ds \dfrac {3^n - 1} {2^n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\frac {3^n} {2^n} - \frac 1 {2^n} } {1 + \frac 1 {2^n} }\) | multiplying top and bottom by $\dfrac 1 {2^n}$ | |||||||||||
\(\ds \) | \(\to\) | \(\ds \dfrac {3^n} {2^n}\) | as $n \to \infty$ |
Thus:
\(\ds \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }\) | \(=\) | \(\ds \cmod {\dfrac {3^{n-1} / 2^{n-1} } {3^n / 2^n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {3 / 2}\) | multiplying top and bottom by $3^{n-1} / 2^{n-1}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 3\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $3 \ \text {(ii)}$