Radius of Convergence of Power Series in Complex Plane

Theorem
Consider the complex power series:
 * $S = \displaystyle \sum_{k \mathop = 0}^\infty z^n$

The radius of convergence $S$ is $1$.

Proof
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.