Exercise 9.  Consider the function  ,  where  .  

9 (a).  Show that    converges uniformly on the set  .  

Solution 9 (a).

See text and/or instructor's solution manual.

Solution.   For  ,  we have   

                

Since  ,  we have   ,   so that   .  

Thus, for    we have

                    ,   

The series    

                    ,  

is known to be convergent  (because    is convergent when  ).  

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

We are done.   

Aside.  We can let Mathematica find the sum of the series.

                    The image of   under the mappings    for  .

                    The image of   under the mapping  .  

Remark.  The function    is called the Riemann Zeta function.

For more information, see the book by Milton Abramowitz and Irene A. Stegun, editors.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.  (1972). New York: Dover. ISBN 0-486-61272-4.

This solution is complements of the authors.

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