124 CHAPTER 4 • SEQUENCES, J ULIA AND MANDELBROT SETS, AND POWER SERIESspecify it by writings (1) = z1, s (2) = Z2, and so on. The values z 1 , z 2 , z 3 , ...
are called the terms of a sequence, and mathematicians, generally being lazy
when it comes to such things, often refer to z1, z2, Z3, ... 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 writ ten as {zn}:=l• {zn};'° or, when
the indices are understood, as {zn}· Mathematicians are also not so fussy about
starting a sequence at z1 so that {Zn};:'= -t> { Zk} ~o,... 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 2 + 5i = (2,5), 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
{zn};'° has ( as its limit as n approaches infinity, provided the terms Zn can be
made as close as we want to ( by making n large enough. When this happens,
we write
lim Zn = (, or Zn -+ ( as n -+ oo.
n->oo
If Jim Zn=(, we say that the sequence {zn}~ converges to (.
n-+oo
(4-5)We need a rigorous definition for Statement (4-5), however, if we are to do
honest mathematics.Definition 4.1: Limit of a sequencen -+lim oo Zn = ( means that for any real number E: >^0 there corresponds a positive
integer N, (which depends on E:) such that Zn E D, (() whenever n > N,. That
is, !Zn -(I < E: whenever n > N,.
Remark 4.1 The reason that we use the notation N, is to emphasize the fact
that this number depends on our choice of E:. Sometimes, for convenience, we
drop the subscript. •Figure 4.1 illustrates a convergent sequence.
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.