Exercise 3.  Find the radius of convergence of the following.  

3 (e).  .  

Solution 3 (e).

See text and/or instructor's solution manual.

Answer   The radius of convergence of    is   .

Solution.  For   ,  we have  .  

Use Theorem 4.16 and calculate the limit superior in the Cauchy-Hadamard Formula:    

                    .

The radius of convergence of the series    is  .

Thus     converges for all  .

We are done.   

Aside.  We can split this series up into two familiar geometric series:  

                    .

The first series is

                    ,   and it converges for  .

The second series is

                    ,  and it converges for  .

Therefore, the series is

                    ,  

converges for  .   

Therefore, the radius of convergence of the series    is  .

Thus     converges for all  .

We are really done.   

Aside.  We can let Mathematica double check our work.

                              The domain set    that is used to produce the images under  

     Graphs of the mappings  ,   ,   ,   ,   ,   and   .

This solution is complements of the authors.

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