Exercise 1.  This exercise relates to Figure 7.1.  

                                      Figure 7.1  The geometric series does not converge uniformly on  .  

1 (a).  For  ,  is the graph of    above or below  ?   Explain.

Solution 1 (a).

See text and/or instructor's solution manual.

Solution.   By definition,      so that   .   

It appears from the graph that the value of the upper function    (in red)  is approximately  ,  

(certainly larger than , so the graph of must be above the graph of    (in blue).  

Thus, for  ,  the graph of      is above   .

                                                              The graphs of      and   .

      We will ask for more details about this situation in Exercise 1 (b).

This solution is complements of the authors.

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