Exercise 5.  Why can't we use the arguments of Theorem 7.2 to prove that the geometric series    converges uniformly on all of  ?  

Solution 5.

See text and/or instructor's solution manual.

Solution.   The crucial step in the proof of Theorem 7.2 is the statement, "Moreover, for all    it is clear that  

            .     

Applying Theorem 7.2 we write     and here we have    for all  n,  and  .  

If we set  ,  then     does not converge.

This solution is complements of the authors.

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