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:
 * $a_n = \dfrac {3^n - 1} {2^n + 1}$

The complex power series:


 * $S = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$

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

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

By Radius of Convergence from Limit of Sequence:
 * $R = \displaystyle \lim_{n \mathop \to \infty} \cmod {\dfrac {a_{n - 1} } {a_n} }$

Thus:

Thus: