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 $1$.

Proof
To determine the radius of convergence of $S$, we examine the limit to infinity of:
 * $\displaystyle \lim_{n \mathop \to \infty} \dfrac {\cmod {a_n z^n} } {\cmod {a_{n - 1} z^{n - 1} } }$

Thus:

Thus:

By the Ratio Test, it follows that:


 * $S$ is convergent for $\cmod z < 1$


 * $S$ is divergent for $\cmod z > 1$.

Hence the result by definition of radius of convergence.