# Complex Power Series/Examples/3^n-1 over 2^n+1

## Example of Complex Power Series

Let $\sequence {a_n}$ be the sequence defined as:

$a_n = \dfrac {3^n - 1} {2^n + 1}$
$S = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$

has a radius of convergence of $\dfrac 2 3$.

## Proof

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

$R = \displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }$

Thus:

 $\displaystyle a_n$ $=$ $\displaystyle \dfrac {3^n - 1} {2^n + 1}$ $\displaystyle$ $=$ $\displaystyle \dfrac {\frac {3^n} {2^n} - \frac 1 {2^n} } {1 + \frac 1 {2^n} }$ multiplying top and bottom by $\dfrac 1 {2^n}$ $\displaystyle$ $\to$ $\displaystyle \dfrac {3^n} {2^n}$ as $n \to \infty$

Thus:

 $\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }$ $=$ $\displaystyle \cmod {\dfrac {3^{n-1} / 2^{n-1} } {3^n / 2^n} }$ $\displaystyle$ $=$ $\displaystyle \dfrac 1 {3 / 2}$ multiplying top and bottom by $3^{n-1} / 2^{n-1}$ $\displaystyle$ $=$ $\displaystyle \frac 2 3$

$\blacksquare$