Solution 7.

Answer.        valid for   .   

Solution.   Use the known series   

                    ,    and   

                    .  

Then

                        

We are done.   

Aside.  We can let Mathematica double check our work.

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