Corollary 4.2. If  ,  the series   converges to .  That is, if then  

            ,  
or equivalently,
            .  

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