Solution 9 (b).

Answer.   Series I.       for    ,   

Answer.   Series II.      for    .

Solution.   Second series.

Start by writing the function in the form

                    .  

Series II.   Differentiate the geometric series      which is valid for     to establish the identity  

                        which is valid for   .  

Make the substitution    and get:

                       
                    

Therefore,        is valid for   .  

We are done.   

Aside.  We can let Mathematica double check our work.

The second 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 image of      under the mapping     for  .

                    The image of      under the mapping   .  

Recall Exercise 8.   Differentiate the geometric series      which is valid for     to establish the identity  

                        which is valid for   .  

The details are similar to those given in Exercise 8.

                       

This solution is complements of the authors.

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