Theorem 4.12 (Geometric Series). If  ,  the series    converges to  .  That is, if    then  

(4-11)            .  

If  ,  the series diverges.  

Exploration.

We can use Mathematica to plot the images of the disk     under the mappings  .  

Hence we can view how the complex geometric series converges to its limit function .

(c) 2006 John H. Mathews, Russell W. Howell