Exercise 9.  Prove Theorem 4.13 (d'Alembert's Ratio Test).  If    is a complex series with the property that  

            ,  

then the series converges absolutely if    and diverges if  .  

Solution 9.

See text and/or instructor's solution manual.

Answer.   Mimic the argument most calculus texts give for real series, but replace |x| with |z|.

Solution.   Assume    exists.

First, suppose that  .

We can select a number  r  such that  .

There exists a positive integer  N  such that for all    we have   ,   this means that

                    ,  

                    ,  

                    ,  

and in general

                    .

Since  ,  Theorem 4.12 implies that      converges.

Since is a positive constant we can conclude that     converge.

Since    for  ,  we have  

                    ,  

and Theorem 4.8 implies that    converges,

and then Corollary 4.1 implies that    converges,  and equivalently    converges,

We can add a finite number of terms to this and conclude that     converges.

The proof of the first part is now complete.

        For the second part,  ,  and suppose that  .

We can select a number  R  such that  .

There exists a positive integer  N  such that for all    we have  ,   this means that

                     

and it follows that  .   

In Exercise 17 in Section 4.1 we asked you to prove:      iff   .  

Therefore, we can conclude that   .

In Exercise 9 in Section 4.1 we asked you to prove:   If      converges, then   .   

Therefore, we can conclude that     diverges.                    Q. E. D.

This solution is complements of the authors.

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