Exercise 5.  Use the ratio test to find a disk in which the following geometric series converge.  

5 (a).  .

Solution 5 (a).

See text and/or instructor's solution manual.

Answer.   Converges in   .

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

                      

When   ,   the series    will converge.  

Solve     and obtain the disk  ,  which can be written as  .

We are done.   

Aside.  The sum of this geometric series is  .  

Notice that    is not defined at the point  ,  where the denominator is zero.  

The distance from    to the center    of the geometric series is  

                    

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   .

Remark 1.  In Section 2.1 we saw that the image of a circle under a "linear mapping" is a circle, and in Section 2.5 we saw that the image of a "circle" under a the reciprocal transformation is a "circle."  Here we have    and it is the composition of a linear mapping followed by the reciprocal mapping, it too will map "circles" onto "circles".  So it is not surprising that the final graph is a circle.  In Section 10.2 we will see that the bilinear transformation or Möbius transformation also has this property.
Remark 2.  It is worthwhile to compare these images with the ones in 5 (b), (c), (d).

Comparison of Geometric Series
Solution.

This solution is complements of the authors.

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