4.3  Geometric Series and Convergence Theorems

    We begin this section by presenting a series of the form   ,  which is called a geometric series and is one of the most important series in mathematics.

Real Exploration.

        Figure 4.A  The image of the disk    under the mapping  .  

Theorem 4.12 (Geometric Series). If  ,  the series    converges to  .  That is, if    then  

(4-11)            .  

If  ,  the series diverges.  

Proof.

Exploration.

Corollary 4.2. If  ,  the series   converges to .  That is, if then  

            ,  
or equivalently,
            .  

If  ,  the series diverges.

Proof.

        Figure 4.B  The annulus of the disk    under the mapping  .  

 Exploration.

Corollary 4.3. If  ,  then for all n,  

            .  

Proof.

Exploration.

Example 4.13.  Show that  .  

Solution.  If we set  ,  then  .  By Theorem 4.12, the sum is  

            .

Explore Solution 4.13.

Example 4.14.  Evaluate  .  

Solution.  We can put this expression in the form of a geometric series:  
            
              

Explore Solution 4.14.

Remark 4.3.  The equality given in Example 4.14 illustrates an important point when evaluating a geometric series whose beginning index is other than zero. The value of equals. If we think of z as the "ratio'' by which a given term of the series is multiplied to generate successive terms, we see that the sum of a geometric series equals , provided  .  

    The geometric series is used in the proof of Theorem 4.12, which is known as the ratio test. It is one of the most commonly used tests for determining the convergence or divergence of series.  The proof is similar to the one used for real series, and we leave it for you to do.

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  .  

Proof.

Example 4.15.  Show that    converges.

Solution.  Using the ratio test, we find that  

               

Because  ,  the series converges.

Explore Solution 4.15.

Example 4.16.  Show that   converges for all  z  in the disk  .  

Solution.  Using the ratio test, we find that  

            .

If  ,  then  ,  and the series converges.  

If  ,  then  ,  and the series diverges.

Explore Solution 4.16.

    Our next result, known as the root test, is slightly more powerful than the ratio test. Before we present this test, we need to discuss a rather sophisticated idea used with it-the limit supremum.

Definition 4.10 (Limit Supremum).  Let    be a sequence of positive real numbers.  The limit supremum of the sequence  (denoted by )  is the smallest real number  L  with the property that for any    there are at most finitely many terms in the sequence that are larger than  .  If there is no such number  L,  then we set  .  

Example 4.17.  The limit supremum of the sequence    is   ,   because if we set  ,  then for any  ,  there are only finitely many terms in the sequence larger than  .  Additionally, if L is smaller than 5, then by setting  ,  we can find infinitely many terms in the sequence larger than    (because  ).

Explore Solution 4.17.

Example 4.18.  The limit supremum of the sequence   is   ,   because if we set  ,  then for any  ,  there are only finitely many terms (actually, there are none) in the sequence larger than  .  Additionally, if L is smaller than 3, then by setting    we can find infinitely many terms in the sequence larger than  ,  because  ,  as the following calculation shows:  

            .  

Explore Solution 4.18.

Example 4.19.  The limit supremum of the Fibonacci sequence    is   .  

(The Fibonacci sequence satisfies the relation    for  ).  

Explore Solution 4.19.

    The limit supremum is a powerful idea because the limit supremum of a sequence always exists, which is not true for the ordinary limit.  However, Example 4.20 illustrates the fact that, if the limit of a sequence does exist, then it will be the same as the limit supremum.

Example 4.20.  The sequence    has  .  
We leave verification of this as an exercise.

Explore Solution 4.20.

Theorem 4.14 (Root Test).  Suppose that the series  ,  has     (i.e.  ).  
Then the series is absolutely convergent if    and divergent if  .  

Proof.

    Note that in applying either Theorems 4.13 or 4.14, if    the convergence or divergence of the series is unknown,  and further analysis is required to determine the true state of affairs.

Extra Example 1.  Find the radius of convergence of the infinite series  .

Explore Extra Solution 1.

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Geometric Series

Convergence of Series

Power Series

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(c) 2006 John H. Mathews, Russell W. Howell