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