Exercise 3.  Is the series      convergent?   Why or why not?  

Solution 3.

See text and/or instructor's solution manual.

Answer.   The series converges by the ratio test.   

Solution   Write    and use d'Alembert's ratio test and calculate the limit value  L:  

                    

Since   ,   the series      will converge.  

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 5.1 we will find that complex series are extensions of real series and we will derive the formula

                    ,

which converges for all  z.

This will give rise to the computation  

                       

Aside.  We can let Mathematica double check our work.

                    A few of the partial sums  ,  and the limit point  .

This solution is complements of the authors.

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