1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 175

Definition 4.4 Sequences that converge to a finite limit are called con-

vergent sequences. Sequences that fail to converge to a finite limit are said

to be divergent.

Two types of divergent sequences may be distinguished: those that are

unbounded, i.e., such that lznl > M for n large enough, whatever be the

given positive number M, and those that although remaining bounded, do

not tend to any limit. In the first case we say that the sequence diverges to

infinity, and write Zn --+ oo (compare with Definition 3.4). In the second

case we say that the sequence is oscillating.

We point out that the foregoing concepts are relative to the metric space

(C, d), where d stands for the Euclidean distance. If in Definition 4.3

the restriction on L to be finite is removed, and the inequality (4.2-1) is

written in terms of the chordal distance, namely as x( Zn, L) < i:, then in

( C*, x) there are no unbounded sequences, and a sequence that diverges to

oo according to Definition 4.4 will be said to .be convergent to oo in the

chordal metric of C* (see Example 2 below).

Examples 1. Let Zn = (1/n) + [(n + l)/n]i. Then limzn = i. In fact,
for any given E > 0 we have lzn - ii = 1(1 + i)/nl = ./2/n < E as soon as
n > ./2/ E. Hence the given sequence is convergent.

2. For Zn = n^2 i we have lznl = n^2 > M whenever n > M^112. Thus
   the sequence diverges to oo (in the sense of Definition 4.4). However, in
   (C, x) we have x(n^2 i, 00) = 1/(1 + n^4 )^112 < t by taking n > (c^2 - 1)^114 ,
   and the sequence converges to oo in the space (C, x).

3. Consider the sequence defined by Zn= (1/n)+(-lri. This sequence

is bounded but does not approach any finite limit. The odd-numbered
terms approach -i, while the even-numbered terms approach +i. The
sequence diverges and belongs to the oscillating type.

4.3 Properties of Convergent Sequences

Convergent sequences have a number of properties similar to those stated
in Theorems 3.1 and 3.2, which is not surprising in view of our remarks
following Definition 4.3. ··Other properties refer specifically to sequences.
For convenience we summarize all these properties in the following theorem.

Theorem 4.1 The following properties of convergent sequences hold:

1. If a sequence {zn} converges, its limit is unique. In other words, a

convergent sequence determines its limit uniquely.

2. A convergent sequence is bounded, and if Zn --+ L and lzn I :::; K for
   n > N, then ILi :::; K.

3. If Zn --+ L, then every subsequence { Znk} also converges to L.