Solution 1 (a).

Answer.   Series I.       is valid for   .   

Answer.   Series II.      is valid for   .   

Solution.      First series.

Series I.   Use the geometric series      which is valid for      and get:

                        

Therefore,      is valid for  .   

We are done.   

Aside.  We can let Mathematica double check our work.

The first series.

We are really done.   

Aside.  As an optional experiment, we can plot some of the partial sums of the series   .  

The mapping    is not one-to-one in the punctured disk  ,

and so we will choose the small portion  .  

                    The images of    under    for  .  

                    The image of      under the mapping   .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell