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

3 (a).      converge uniformly on   .  

Solution 3 (a).

See text and/or instructor's solution manual.

Solution.   Here   ,   and  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 limit function.

We are really done.   

Remark.  The function    is differentiable and

                    .

The    function is sometimes called the dilogarithm.  It is useful in theoretical physics.

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.

We are really really done.   

                    The images of   under the mappings    for  .

                    The image of   under the mapping  .  

This solution is complements of the authors.

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