Exercise 1.  This exercise relates to Figure 7.1.  

                                      Figure 7.1  The geometric series does not converge uniformly on  .  

1 (c).  Assuming that the graph is accurate to scale, what is the value of  n  in  ?  Explain.

Solution 1 (c).

See text and/or instructor's solution manual.

Solution.   From the graph, we approximate  .   Using   ,  

we observe that      and deduce that  .  

                    The graphs of       and   .
                    It is revealed that   .

We are done.   

Aside.  The partial sum    looks close   over the interval  .

Let us look at the image of the disk    under the mappings  ,  for  .  

                    The images of the disk   under  ,  for  .  

                    The image of the disk   under the mapping  .   

Aside.  In Section 10.2 we will learn that the image of the disk    is the disk  .

This solution is complements of the authors.

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