**for**

**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:

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 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 .

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

The sequence of points converges to .

**Example 4.2.** Show
that the sequence diverges.

Solution. We have

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

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

The sequence of points diverges.

**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.

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 of Theorem 4.3 is in the book.
**

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 of Theorem 4.4 is in the book.
**

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

**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.

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

The partial sums converge to the value .

**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.

**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.

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

The sequence of partial sums diverges.

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

and

.

**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 of Theorem 4.8 is in the book.
**

**Corollary 4.1. **
If converges,
then converges.

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

**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 .

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

The partial sums converge to the value

**Exercises
Section 4.1. Sequences and
Series**** **

**The Next Module
is**

**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