Exercise 7.  Suppose the function    has radius of convergence  .

Verify part (ii)  for all  k,     of Theorem 4.17 for all  k  by using mathematical induction.  

Solution 7.

See text and/or instructor's solution manual.

Solution.   This exercise builds on ideas that were investigated in Exercise 5 of Section 3.1.

Part (i) of Theorem 4.17 establishes     when  k=1.  

Assume   is true for some , and set    or    then this series can be written as  

                    

                    
                    
                    

                    
                    
where  .

        In other words,   .

Applying part (i) of Theorem 4.17 to the function gives  

                      

Now set    or    then this series can be written as

                    

                    

                    

which is what we needed to establish.

We are done.   

Alternate solution.  Let   denote the binomial coefficient,  i. e.  .  

Then the function    can be written as

                    ,
          
then the k-th derivative is

                    .
          
This representation can be found in advanced books.  

This solution is complements of the authors.

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