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).