Module

for

Complex Sequences and Series

 

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:  

[Graphics:Images/ComplexSequenceSeriesMod_gr_1.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_2.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_3.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_4.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_5.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_6.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_7.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_8.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_9.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_10.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_11.gif]

[Graphics:Images/ComplexSequenceSeriesMod_gr_12.gif]


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  [Graphics:Images/ComplexSequenceSeriesMod_gr_21.gif],  and so on.  The values   [Graphics:Images/ComplexSequenceSeriesMod_gr_22.gif],  are called the terms of a sequence, and mathematicians, being generally lazy when it comes to such things, often refer to [Graphics:Images/ComplexSequenceSeriesMod_gr_23.gif] 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  [Graphics:Images/ComplexSequenceSeriesMod_gr_24.gif], [Graphics:Images/ComplexSequenceSeriesMod_gr_25.gif], or when the indices are understood, as [Graphics:Images/ComplexSequenceSeriesMod_gr_26.gif].  Mathematicians are also not so fussy about starting a sequence at  [Graphics:Images/ComplexSequenceSeriesMod_gr_27.gif], so that  [Graphics:Images/ComplexSequenceSeriesMod_gr_28.gif], [Graphics:Images/ComplexSequenceSeriesMod_gr_29.gif], 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:  

    [Graphics:Images/ComplexSequenceSeriesMod_gr_30.gif],  [Graphics:Images/ComplexSequenceSeriesMod_gr_31.gif],  [Graphics:Images/ComplexSequenceSeriesMod_gr_32.gif],  [Graphics:Images/ComplexSequenceSeriesMod_gr_33.gif], [Graphics:Images/ComplexSequenceSeriesMod_gr_34.gif], ...

 

    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  [Graphics:Images/ComplexSequenceSeriesMod_gr_35.gif],  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  [Graphics:Images/ComplexSequenceSeriesMod_gr_36.gif]  has  [Graphics:Images/ComplexSequenceSeriesMod_gr_37.gif]  as its limit as  n  approaches infinity, provided the terms  [Graphics:Images/ComplexSequenceSeriesMod_gr_38.gif]  can be made as close as we want to  [Graphics:Images/ComplexSequenceSeriesMod_gr_39.gif]  by making  n  large enough.  When this happens, we write

(4-5)        [Graphics:Images/ComplexSequenceSeriesMod_gr_40.gif].

If  [Graphics:Images/ComplexSequenceSeriesMod_gr_41.gif],  we say that the sequence  [Graphics:Images/ComplexSequenceSeriesMod_gr_42.gif]  converges to  [Graphics:Images/ComplexSequenceSeriesMod_gr_43.gif].  

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

 

Definition 4.1 (Limit of a Sequence).  [Graphics:Images/ComplexSequenceSeriesMod_gr_44.gif]  means that for any real number  [Graphics:Images/ComplexSequenceSeriesMod_gr_45.gif]  there corresponds a positive integer  [Graphics:Images/ComplexSequenceSeriesMod_gr_46.gif]  (which depends on  [Graphics:Images/ComplexSequenceSeriesMod_gr_47.gif])  such that  [Graphics:Images/ComplexSequenceSeriesMod_gr_48.gif]  whenever  [Graphics:Images/ComplexSequenceSeriesMod_gr_49.gif].   That is  [Graphics:Images/ComplexSequenceSeriesMod_gr_50.gif]  whenever  [Graphics:Images/ComplexSequenceSeriesMod_gr_51.gif].  Figure 4.1 illustrates a convergent sequence.

[Graphics:Images/ComplexSequenceSeriesMod_gr_52.gif]

            Figure 4.1   A sequence [Graphics:Images/ComplexSequenceSeriesMod_gr_53.gif] that converges to [Graphics:Images/ComplexSequenceSeriesMod_gr_54.gif].   (If  [Graphics:Images/ComplexSequenceSeriesMod_gr_55.gif] then [Graphics:Images/ComplexSequenceSeriesMod_gr_56.gif].)

 

Remark 4.1. The reason we use the notation [Graphics:Images/ComplexSequenceSeriesMod_gr_57.gif] is to emphasize the fact that this number depends on our choice of [Graphics:Images/ComplexSequenceSeriesMod_gr_58.gif].  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  [Graphics:Images/ComplexSequenceSeriesMod_gr_59.gif]  and  [Graphics:Images/ComplexSequenceSeriesMod_gr_60.gif].  Then

(4-6)            [Graphics:Images/ComplexSequenceSeriesMod_gr_61.gif],   iff  

(4-7)            [Graphics:Images/ComplexSequenceSeriesMod_gr_62.gif].  

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  [Graphics:Images/ComplexSequenceSeriesMod_gr_63.gif].  

Solution.  We write  [Graphics:Images/ComplexSequenceSeriesMod_gr_64.gif].  Using results concerning sequences of real numbers, we find that  

    [Graphics:Images/ComplexSequenceSeriesMod_gr_65.gif]    and    [Graphics:Images/ComplexSequenceSeriesMod_gr_66.gif].  

Therefore  [Graphics:Images/ComplexSequenceSeriesMod_gr_67.gif].  

Aside.  Just for fun, we can graph some of the terms in this complex sequence.   

                         [Graphics:Images/ComplexSequenceSeriesMod.0_gr_1.gif]   

                    The sequence of points  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_2.gif]  converges to  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_3.gif].

Explore Solution 4.1.

 

Example 4.2.  Show that the sequence  [Graphics:Images/ComplexSequenceSeriesMod_gr_82.gif]  diverges.  

Solution.  We have  

            [Graphics:Images/ComplexSequenceSeriesMod_gr_83.gif]  

The real sequences  [Graphics:Images/ComplexSequenceSeriesMod_gr_84.gif]  and  [Graphics:Images/ComplexSequenceSeriesMod_gr_85.gif]  both exhibit divergent oscillations, so we conclude that  [Graphics:Images/ComplexSequenceSeriesMod_gr_86.gif]  diverges.

Aside.  Just for fun, we can graph some of the terms in this divergent complex sequence.   

                    [Graphics:Images/ComplexSequenceSeriesMod.0_gr_4.gif]    

                    The sequence of points  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_5.gif]  diverges.

Explore Solution 4.2.

 

Definition 4.2 (Bounded Sequence).  A complex sequence [Graphics:Images/ComplexSequenceSeriesMod_gr_105.gif] is bounded provided that there exists a positive real number R and an integer N such that  [Graphics:Images/ComplexSequenceSeriesMod_gr_106.gif]  for all  [Graphics:Images/ComplexSequenceSeriesMod_gr_107.gif].  In other words, for [Graphics:Images/ComplexSequenceSeriesMod_gr_108.gif], the sequence [Graphics:Images/ComplexSequenceSeriesMod_gr_109.gif] is contained in the disk  [Graphics:Images/ComplexSequenceSeriesMod_gr_110.gif].  

 

    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 [Graphics:Images/ComplexSequenceSeriesMod_gr_111.gif] is a convergent sequence, then [Graphics:Images/ComplexSequenceSeriesMod_gr_112.gif] is bounded.  

Proof.

 

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

 

Definition 4.3 (Cauchy Sequence).  The sequence [Graphics:Images/ComplexSequenceSeriesMod_gr_113.gif] is said to be a Cauchy sequence if for every  [Graphics:Images/ComplexSequenceSeriesMod_gr_114.gif]  there exists a positive integer  [Graphics:Images/ComplexSequenceSeriesMod_gr_115.gif],  such that if  [Graphics:Images/ComplexSequenceSeriesMod_gr_116.gif],  then  [Graphics:Images/ComplexSequenceSeriesMod_gr_117.gif],  or, equivalently,  [Graphics:Images/ComplexSequenceSeriesMod_gr_118.gif].  

 

    The following should now come as no surprise.

 

Theorem 4.3,  (Cauchy Sequences Converge).  If  [Graphics:Images/ComplexSequenceSeriesMod_gr_119.gif]  is a Cauchy sequence, then [Graphics:Images/ComplexSequenceSeriesMod_gr_120.gif]  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 [Graphics:Images/ComplexSequenceSeriesMod_gr_121.gif], and form a new sequence [Graphics:Images/ComplexSequenceSeriesMod_gr_122.gif], called the sequence of partial sums, as follows.

                [Graphics:Images/ComplexSequenceSeriesMod_gr_123.gif]   

 

Definition 4.4 (Infinite Series).  The formal expression [Graphics:Images/ComplexSequenceSeriesMod_gr_124.gif]  is called an infinite series, and  [Graphics:Images/ComplexSequenceSeriesMod_gr_125.gif],  are called the terms of the series.  

 

    If there is a complex number S for which

            [Graphics:Images/ComplexSequenceSeriesMod_gr_126.gif],  
            
we will say that the infinite series [Graphics:Images/ComplexSequenceSeriesMod_gr_127.gif] converges to S, and that S is the sum of the infinite series.  When this occurs, we write

            [Graphics:Images/ComplexSequenceSeriesMod_gr_128.gif].  

    The series  [Graphics:Images/ComplexSequenceSeriesMod_gr_129.gif]  is said to be absolutely convergent provided that the (real) series of magnitudes  [Graphics:Images/ComplexSequenceSeriesMod_gr_130.gif]  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  [Graphics:Images/ComplexSequenceSeriesMod_gr_131.gif]  converges, then so does  [Graphics:Images/ComplexSequenceSeriesMod_gr_132.gif],  where  [Graphics:Images/ComplexSequenceSeriesMod_gr_133.gif]  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  [Graphics:Images/ComplexSequenceSeriesMod_gr_134.gif]  and  [Graphics:Images/ComplexSequenceSeriesMod_gr_135.gif].  Then  

            [Graphics:Images/ComplexSequenceSeriesMod_gr_136.gif]     (converges)

if and only if both  

            [Graphics:Images/ComplexSequenceSeriesMod_gr_137.gif]     (converge).  

Proof.

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

 

Theorem 4.5.  If  [Graphics:Images/ComplexSequenceSeriesMod_gr_138.gif]  is a convergent complex series, then  [Graphics:Images/ComplexSequenceSeriesMod_gr_139.gif].  

Proof.

 

Example 4.3.  Show that the series  [Graphics:Images/ComplexSequenceSeriesMod_gr_140.gif]  [Graphics:Images/ComplexSequenceSeriesMod_gr_141.gif]is convergent.

Solution.  Recall that the real series  [Graphics:Images/ComplexSequenceSeriesMod_gr_142.gif]  and  [Graphics:Images/ComplexSequenceSeriesMod_gr_143.gif]  are convergent.  Hence, Theorem 4.4 implies that the given complex series is convergent.  

Aside.  Just for fun, we can graph some of the partial sums of this complex series.   

                              [Graphics:Images/ComplexSequenceSeriesMod.0_gr_6.gif]   

                    The partial sums  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_7.gif]  converge to the value  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_8.gif].

Explore Solution 4.3.

 

Example 4.4.  Show that the series  [Graphics:Images/ComplexSequenceSeriesMod_gr_155.gif]  is divergent.

Solution.  We know that the real series  [Graphics:Images/ComplexSequenceSeriesMod_gr_156.gif]  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 [Graphics:Images/ComplexSequenceSeriesMod_gr_174.gif] is divergent.  

Solution.  Here we set  [Graphics:Images/ComplexSequenceSeriesMod_gr_175.gif]  and observe that  

            [Graphics:Images/ComplexSequenceSeriesMod_gr_176.gif].  

Thus [Graphics:Images/ComplexSequenceSeriesMod_gr_177.gif], and Theorem 4.5 implies that the series is not convergent;  hence it is divergent.

Aside.  Just for fun, we can graph some of the partial sums of this divergent complex series.   

                    [Graphics:Images/ComplexSequenceSeriesMod.0_gr_9.gif]  

                    The sequence of partial sums  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_10.gif]  diverges.

Explore Solution 4.5.

 

Theorem 4.6.  Let  [Graphics:Images/ComplexSequenceSeriesMod_gr_253.gif]  be convergent series, and let  c  be a complex number.  Then  

            [Graphics:Images/ComplexSequenceSeriesMod_gr_254.gif]  
        and  
            [Graphics:Images/ComplexSequenceSeriesMod_gr_255.gif].  

Proof.

 

Definition 4.5 (Cauchy Product of Series).  Let  [Graphics:Images/ComplexSequenceSeriesMod_gr_256.gif]  and  [Graphics:Images/ComplexSequenceSeriesMod_gr_257.gif]  be convergent series, where [Graphics:Images/ComplexSequenceSeriesMod_gr_258.gif]  are complex numbers.  The Cauchy product of the two series is defined to be the series  [Graphics:Images/ComplexSequenceSeriesMod_gr_259.gif],  where  [Graphics:Images/ComplexSequenceSeriesMod_gr_260.gif].  

 

Theorem 4.7.  If the Cauchy product converges, then  

            [Graphics:Images/ComplexSequenceSeriesMod_gr_261.gif],  
        where  
            [Graphics:Images/ComplexSequenceSeriesMod_gr_262.gif]

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  [Graphics:Images/ComplexSequenceSeriesMod_gr_263.gif]  be a convergent series of real nonnegative terms.  If  [Graphics:Images/ComplexSequenceSeriesMod_gr_264.gif]  is a sequence of complex numbers and  [Graphics:Images/ComplexSequenceSeriesMod_gr_265.gif]  holds for all  n, then  [Graphics:Images/ComplexSequenceSeriesMod_gr_266.gif]  converges.

Proof.

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

 

Corollary 4.1. If    [Graphics:Images/ComplexSequenceSeriesMod_gr_267.gif]   converges, then   [Graphics:Images/ComplexSequenceSeriesMod_gr_268.gif]   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  [Graphics:Images/ComplexSequenceSeriesMod_gr_269.gif]  is convergent.

Solution.  We calculate  [Graphics:Images/ComplexSequenceSeriesMod_gr_270.gif].    Using the comparison test and the fact that  [Graphics:Images/ComplexSequenceSeriesMod_gr_271.gif]  converges, we determine that  [Graphics:Images/ComplexSequenceSeriesMod_gr_272.gif]  converges and hence, by Corollary 4.1, so does [Graphics:Images/ComplexSequenceSeriesMod_gr_273.gif].

Aside.  Just for fun, we can graph some of the partial sums of this complex series.   

                              [Graphics:Images/ComplexSequenceSeriesMod.0_gr_11.gif]  

                    The partial sums  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_12.gif]  converge to the value  [Graphics:Images/ComplexSequenceSeriesMod.0_gr_13.gif]

Explore Solution 4.6.

 

Exercises Section 4.1.  Sequences and Series

 

Library Research Experience for Undergraduates

Geometric Series

Convergence of Series

Power Series

 

 

 

 

The Next Module is

Julia and Mandelbrot Sets

 

 

 

Return to the Complex Analysis Modules  

 

 

 

Return to the Complex Analysis Project

 

 

  

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell