174 Chapter4
Occasionally, we need to consider functions from the set of all integers
I into C. These functions, called two-sided infinite sequences, are denoted
... , Z-2, Z-1, zo, z1, z2, ... or briefly, {zn}~:.
As in section 2.2, the elements z 1 , z 2 , ... of C are called the terms of
the sequence {zn}, and Zn is called the general term. The range off is said
to be the range of the sequence, denoted {zn} *. It is essential to bear in
mind the distinction between a sequence and its range.
For instance, if Un= (-l)n (n = 1, 2, ... ), the sequence is -1, 1, -1, ... ,
while {un}* = {-1,1}. This example shows that the range of an in£.nite
sequence could be a finite set.
If M = {n1,n2,n3, ... } is a subset of J such that ni < n2 < n3 < ·· ·,
the sequence {znk}~ = Znp Zn 2 , ••• is called a subsequence of {zn}. This
is really a composite function g o f, where f: J ---+ M and g: M ---+ <C.
A sequence of complex numbers induces (by composition) various other
sequences of real numbers, namely, {Rezn} = {xn}, {Imzn} = {Yn},
{lznl} = {rn}, {Argzn} = {Bn}, ... , as well as some related sequences
of complex numbers, such as {zn}, and more generally, {wn} = {f(zn)}.
If F denotes the family of all complex functions f of a single variable,
a sequence g: J ---+ F is called a sequence of complex functions (or, a
functional sequence), written Un(z)}: for example, {zn}.
Sequences of functions of two or more complex variables are defined
similarly. Sequences of functions are discussed in Section 4.10 and in
Chapter 8.
4.2 Convergence of Sequences
Definition 4.3 Let {zn} be a sequence of complex numbers. We say that
{Zn} has limit L ( L -:j; oo ), or tends to L, or converges to L, as n ---+ oo, iff
to every € > 0 there corresponds a positive number N, such that n > N
implies that
or Zn E N<(L) (4.2-1)We write limn--+oo Zn = L or Zn ---+ L as n ---+ oo.
A sequence that converges to zero is called a null sequence. Clearly,
if Zn ---+ L, the sequence defined by Un = Zn - L is a null sequence, and
conversely, if Un ---+ 0, then Zn ---+ L.
The concept of the limit of a sequence may be thought as a special
case of the notion of finite limit of a function at infinity (Definition 3.2).
Alternatively, it may be reduced to the case of a finite limit at a finite point
(Definition 3.1). To see this, consider the set S = {1,^1 / 2 , 1/s, ... , l/n, ... }