Exercises Section 4.4.  Power Series Functions

Exercise 1.  Prove part (iii) d'Alembert's Ratio Test of Theorem 4.16 .  

Exercise 2.  Consider the series   ,    ,   and   .  

2 (a).  Show that each series has radius of convergence  .

2 (b).  Show that the first series    converges nowhere on  .  

2 (c).  Show that the second series    converges everywhere on  .

2 (d).  It turns out that the third series    converges everywhere on  ,  except at the point  .  

This is not easy to prove.  Give it a look.

Exercise 3.  Find the radius of convergence of the following.  

3 (a).  .  

3 (b).  .  

3 (c).  .  

3 (d).  .  

3 (e).  .  

3 (f).  .  

3 (g).  .  

3 (h).  .  

3 (i).  .  

3 (j).  .   

Hint:   .

Exercise 4.  Show that   .   For what values of z is this valid?  

Exercise 5.  Suppose that      has radius of convergence  .  

Show that      has radius of convergence  .  

Exercise 6.  Does there exist a power series      that converges at    and diverges at  ?   Why or why not?  

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.  

Exercise 8.  This exercise establishes that the radius of convergence for    given in Theorem 4.17,

is the same as that of the function  .  

8 (a).  Explain why the radius of convergence for g(z)  is   .

8 (b).  Show that  .

Hint: The limit superior equals the limit.  Show that  .  

8 (c).  Assuming that  ,  show that the conclusion for this exercise follows.

8 (d).  Verify the truth of the assumption made in part (c).

Exercise 9.  Here we establish the validity of Inequality (4-17) in the proof of Theorem 4.17.  

9 (a).  Show that

where s and t are arbitrary complex numbers,  .  

9 (b).  Explain why, in Inequality (4-17) ,     and  .  

9 (c).  Let    and    in part (a) to establish inequality  (4-17).  

Exercise 10.  Show that the radius of convergence of the series for

          ,   and   ,   is   .  

Exercise 11.  Consider the series obtained by substituting for the complex number z the real number x in the Maclaurin series for  .  

Where does this series converge?  

Exercise 12.  Show that,      for   .

Hint:  .  Now use Theorem 4.12.

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