4 Sequences and Series
4.1 Sequences of Complex Numbers
Sequences in general have been defined in Section 2.2, Definition 2.3. In
this section we discuss in some detail sequences with range in C.
Definition 4.1 A finite complex sequence is a single-valued function f
from the set JN = {1, 2, ... , N} of the first N positive integers into the
complex plane, i.e., f: JN --t C. If we let J(n) =Zn for n E JN, the finite
sequence is usually denoted by z1, z2, ... , z N or, briefly, by {Zn}~.
Definition 4.2 An infinite complex sequence is a single-valued function f
from the set J = {1, 2, ... , n, ... } of the positive integers into the complex
plane, i.e., f: J --t C. If, as before, we let f(n) =Zn, n E J, the infinite
sequence is denoted by z1, z2, ... , Zn,... or, briefly, {zn} ;"'.
Unless otherwise stated, in what follows the term sequence will be used
instead of the more precise infinite complex sequence, since we shall be
concerned mostly with this type of sequence, finite sequences being rarely
met in our subsequent work, and the notation will be further abbreviated
by writing {zn}·
At times it is convenient to replace J in Definition 4.2 by the set of all
nonnegative integers, thus beginning with zero rather than with 1. Then
the sequence is written zo, z1, z2, ... , Zn, ... , or {zn}:.
173