1.2. Convergence in C 9
this section is to extend these ideas and results to sequences and series of complex
numbers and power series in a complex variable. This is an introductory section.
A deeper study of complex power series will come later in the text.
There is no natural order relation on the complex numbers. The statement
z w makes no sense for complex numbers z and w. However, if these numbers
happen to be real, then the inequality does make sense, because R is an ordered
field. We make heavy use of inequalities in this section, but note they are always
inequalities between real numbers. Thus, if an inequality of the form a b occurs
in this text the numbers a and b are assumed to be real numbers, even if there is
no explicit statement to that effect.
Convergence of Sequences. A sequence of complex numbers converges if and
only if it converges as a sequence of vectors in
R2.
The formal definition is the
familiar one:
Definition 1.2.1. A sequence {zn} of complex numbers is said to converge to the
number w if, for every 0, there exists an integer N such that
|zn w| whenever n N.
In this case, we say that w is the limit of the sequence zn and write limn→∞ zn = w,
or lim zn = w, or simply zn w.
Remark 1.2.2. There are a couple of simple observations about convergence of
sequences that will prove to be very useful.
(1) If {zn} is a sequence of complex numbers, then lim zn = w if and only if
lim |zn w| = 0.
(2) If {an} and {bn} are sequences of real numbers with 0 an bn and if
bn 0, then also an 0.
These both follow immediately from the definition of limit of a sequence. The
second is one form of what is sometimes called the squeeze principle.
The first of these observations reduces the problem of showing that a sequence
of complex numbers converges to a given complex number to showing that a certain
sequence of non-negative real numbers converges to 0. This is useful because we
have many tools at our disposal to show that a sequence of non-negative numbers
converges to 0. One of the most useful of these tools is the second observation
above. The next example and the proof of the following theorem are excellent
examples of how this works.
Example 1.2.3. Prove that lim
zn
= 0 if |z| 1.
Solution: Note that
|zn|
=
|z|n
follows from Part (e) of Theorem 1.1.7. If
|z| 1 then lim
|z|n
= 0. That lim
zn
= 0, as well, follows from (1) of Remark
1.2.2.
Theorem 1.2.4. A sequence of complex numbers {zn} converges to a complex
number w if and only if {Re(zn)} converges to Re(w) and {Im(zn)} converges to
Im(w).
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