# 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 $1$.

## 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:

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

Thus:

 $\displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }$ $=$ $\displaystyle \cmod {\dfrac {3 / 2} {3 / 2} }$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle 1$ $\quad$ $\quad$

$\blacksquare$