Chapter 4  Sequences, Julia and Mandelbrot Sets, and Power Series

Overview

    In 1980 Benoit Mandelbrot led a team of mathematicians in producing some stunning computer graphics from very simple rules for manipulating complex numbers.  This event marked the beginning of a new branch of mathematics, known as fractal geometry, that has some amazing applications.  Many of the tools needed to appreciate Mandelbrot's work are contained in this chapter.  We look at extensions to the complex domain of sequences and series, ideas that are familiar to students who have completed a standard calculus course.

4.1  Sequences and Series

    In formal terms, a complex sequence is a function whose domain is the positive integers and whose range is a subset of the complex numbers. The following are examples of sequences:  


Exploration

    For convenience, at times we use the term sequence rather than complex sequence.  If we want a function s to represent an arbitrary sequence, we can specify it by writing  ,  and so on.  The values   ,  are called the terms of a sequence, and mathematicians, being generally lazy when it comes to such things, often refer to as the sequence itself, even though they are really speaking of the range of the sequence when they do so.  You will usually see a sequence written as  , , or when the indices are understood, as .  Mathematicians are also not so fussy about starting a sequence at  , so that  , , etc., would also be acceptable notation, provided all terms were defined.  For example, the sequence r given by Equation (4-4) could be written in a variety of ways:  

    ,  ,  ,  , , ...

    The sequences f and g given by Equations (4-1) and (4-2) behave differently as n gets larger.  The terms in Equation (4-1) approach  ,  but those in Equation  (4-2)  do not approach any particular number, as they oscillate around the eight eighth roots of unity on the unit circle.  Informally, the sequence    has    as its limit as  n  approaches infinity, provided the terms    can be made as close as we want to    by making  n  large enough.  When this happens, we write

(4-5)        .

If  ,  we say that the sequence    converges to  .  

    We need a rigorous definition for Statement (4-5), however, if we are to do honest mathematics.

Definition 4.1 (Limit of a Sequence).    means that for any real number    there corresponds a positive integer    (which depends on  )  such that    whenever  .   That is    whenever  .  Figure 4.1 illustrates a convergent sequence.

            Figure 4.1   A sequence that converges to .   (If   then .)

Remark 4.1. The reason we use the notation is to emphasize the fact that this number depends on our choice of .  Sometimes it will be convenient to drop the subscript.

    In form, Definition 4.1 is exactly the same as the corresponding definition for limits of real sequences.  In fact, a simple criterion casts the convergence of complex sequences in terms of the convergence of real sequences.

Theorem 4.1.  Let    and  .  Then

(4-6)            ,   iff  

(4-7)            .  

Proof.

Proof of Theorem 4.1 is in the book.

Complex Analysis for Mathematics and Engineering

Example 4.1.  Find the limit of the sequence  .  

Solution.  We write  .  Using results concerning sequences of real numbers, we find that  

        and    .  

Therefore  .  

Explore Solution 4.1.

Example 4.2.  Show that the sequence    diverges.  

Solution.  We have  

              

The real sequences    and    both exhibit divergent oscillations, so we conclude that    diverges.

Explore Solution 4.2.

Definition 4.2 (Bounded Sequence).  A complex sequence is bounded provided that there exists a positive real number R and an integer N such that    for all  .  In other words, for , the sequence is contained in the disk  .  

    Bounded sequences play an important role in some newer developments in complex analysis that are discussed in Section 4.2.  A theorem from real analysis stipulates that convergent sequences are bounded.  The same result holds for complex sequences.

Theorem 4.2.  If is a convergent sequence, then is bounded.  

Proof.

    As with real numbers, we also have the following definition.  

Definition 4.3 (Cauchy Sequence).  The sequence is said to be a Cauchy sequence if for every    there exists a positive integer  ,  such that if  ,  then  ,  or, equivalently,  .  

    The following should now come as no surprise.

Theorem 4.3,  (Cauchy Sequences Converge).  If    is a Cauchy sequence, then   converges.

Proof.

Proof of Theorem 4.3 is in the book.
Complex Analysis for Mathematics and Engineering

    One of the most important notions in analysis (real or complex) is a theory that allows us to add up infinitely many terms.  To make sense of such an idea we begin with a sequence , and form a new sequence , called the sequence of partial sums, as follows.

                   

Definition 4.4 (Infinite Series).  The formal expression   is called an infinite series, and  ,  are called the terms of the series.  

    If there is a complex number S for which

            ,  
            
we will say that the infinite series converges to S, and that S is the sum of the infinite series.  When this occurs, we write

            .  

    The series    is said to be absolutely convergent provided that the (real) series of magnitudes    converges.  

    If a series does not converge, we say that it diverges.

Remark 4.2.  The first finitely many terms of a series do not affect its convergence or divergence and, in this respect, the beginning index of a series is irrelevant.  Thus, we will without comment conclude that if a series    converges, then so does  ,  where    is any finite collection of terms.  A similar remark holds for determining divergence of a series.

    As you might expect, many of the results concerning real series carry over to complex series.  We now give several of the more standard theorems for complex series, along with examples of how they are used.

Theorem 4.4.  Let    and  .  Then  

                 (converges)

if and only if both  

                 (converge).  

Proof.

Proof of Theorem 4.4 is in the book.
Complex Analysis for Mathematics and Engineering

Theorem 4.5.  If    is a convergent complex series, then  .  

Proof.

Example 4.3.  Show that the series    is convergent.

Solution.  Recall that the real series    and    are convergent.  Hence, Theorem 4.4 implies that the given complex series is convergent.  

Explore Solution 4.3.

Example 4.4.  Show that the series    is divergent.

Solution.  We know that the real series    is divergent.  Hence, Theorem 4.4 implies that the given complex series is divergent.

Explore Solution 4.4.

Example 4.5.  Show that the series is divergent.  

Solution.  Here we set    and observe that  

            .  

Thus , and Theorem 4.5 implies that the series is not convergent;  hence it is divergent.

Explore Solution 4.5.

Theorem 4.6.  Let    be convergent series, and let  c  be a complex number.  Then  

              
        and  
            .  

Proof.

Definition 4.5 (Cauchy Product of Series).  Let    and    be convergent series, where   are complex numbers.  The Cauchy product of the two series is defined to be the series  ,  where  .  

Theorem 4.7.  If the Cauchy product converges, then  

            ,  
        where  
            . 

Proof.

The proof can be found in a number of texts, for example, Infinite Sequences and Series, by Konrad Knopp (translated by Frederick Bagemihl;  New York: Dover, 1956).

Theorem 4.8  (Comparison Test).  Let    be a convergent series of real nonnegative terms.  If    is a sequence of complex numbers and    holds for all  n, then    converges.

Proof.

Proof of Theorem 4.8 is in the book.
Complex Analysis for Mathematics and Engineering

Corollary 4.1. If       converges, then      converges.  

In other words, absolute convergence implies convergence for complex series as well as for real series.

Proof.

Example 4.6.  Show that the series    is convergent.

Solution.  We calculate  .    Using the comparison test and the fact that    converges, we determine that    converges and hence, by Corollary 4.1, so does .

Explore Solution 4.6.

Exercises Section 4.1.  Sequences and Series