Exercise 11.  Consider the series obtained by substituting for the complex number z the real number x in the Maclaurin series for  .  Where does this series converge?  

Solution 11.

See text and/or instructor's solution manual.

Answer.  The series    converges for all values of z by the ratio test, and we will see in Section 5.4 that

                        for all  z.  

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   .  

Thus     converges for all  .

We are done.   

Remark.  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.

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