Solution 5.

Answer.        valid for   .   

Solution.   Use the known series  .  

Make the substitution    and get:

                      

is valid for   .   

We are done.   

Aside.  We can let Mathematica double check our work.

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 region  ,  an 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