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

3 (a).  .  

Solution 3 (a).

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 in in d'Alembert's Ratio Test:    

                       
                    
Since    the radius of convergence of the series    is   .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  The Taylor series for    was studied in calculus, and we have  

                    ,  

which converges for all  x.

In Section 5.4 we will find that complex series are extensions of real series and we will derive the formula

                    ,

which converges for all  z.  Using the substitution   will establish that the series at hand is really

                    .

Remark.  At first glance, a first reaction might be:    could not have a power series expansion about   because does not.

However, the series involves powers of   .

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