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

Solution 1.

See text and/or instructor's solution manual.

Solution.  Given   .  

We must prove that if   and then the series converges for all  .

Recall Theorem 4.13 which we asked you to prove in Exercise 9 in Section 4.3.  If    is a complex series with the property that  

            ,  

then the series converges absolutely if    and diverges if  .  

Set and apply  Theorem 4.13.  

The series    converges absolutely if   ,  and

the series    diverges if   .  

Thus, the series    converges absolutely if   .

If      is finite but not zero, then the series    converges if  .  

If   ,   the series    converges for all  z.

If   ,   the series converges only at the single point  .  

This solution is complements of the authors.

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