Exercise 3.  Prove that the following series converge uniformly on the sets indicated.  

3 (c).      converge uniformly on   ,   where  .  

Solution 3 (c).

See text and/or instructor's solution manual.

Solution.   Here  ,  and  for a fixed  ,  choose    large so that    for all  .  

Then, for all  ,  and all  ,  we have    

                    ,     
                  
so that  

                    .
                    
Recall formula (1-24) in Section 1.3.  

(1-24)           .  

In formula (1-24) we set    and    and get  

                    .

Now use this, together with     and obtain

                    .  

Therefore, it now follows that  

                        

      For    and all  ,  we have    

                    .   

The series    

                      

is known to be convergent  (it is a geometric series).  

Therefore, by the Weierstrass M-test, the series      converges uniformly on   ,   where  .  

                    The image of   under    for  .

                    The image of   under the mapping  .  

This solution is complements of the authors.

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