Complex Analysis - Maths KU

302 Chapter 6 Series and Residues
y
L
ε
Figure 6.1 If {zn} converges to L, all
but a finite number of terms are in
every ε-neighborhood of L.
–1
–
1
5
y
i
4
–
i
2
x
1
3
Figure 6.2 The terms of the sequence
� i n+1 /n � spiral in toward 0.
6.1 Sequences and Series
Much of the theory 6.1of
complex sequences and series is analogous to that encountered in real
calculus.
In this section we explore the definitions of convergence and divergence for complex
sequences and complex infinite series.In addition, we give some tests for convergence of
infinite series.You are urged to pay particular attention to what is said about geometric
series since this type of series will be important in the later sections of this chapter.
x
Sequences A sequence {zn} is a function whose domain is the set of
positive integers and whose range is a subset of the complex numbers C.In
other words, to each integer n = 1, 2, 3, ... we assign a single complex number
zn.For example, the sequence {1+i n } is
1+i, 0, 1 − i, 2, 1+i, . . . .
↑ ↑ ↑ ↑ ↑
n =1, n =2, n =3, n =4, n =5, ...
If lim zn = L, we say the sequence {zn} is convergent. In other words,
n→∞
{zn} converges to the number L if for each positive real number ε an N can
be found such that |zn − L| N.Since |zn − L| is distance,
the terms zn of a sequence that converges to L can be made arbitrarily close
to L.Put another way, when a sequence {zn} converges to L, then all but a
finite number of the terms of the sequence are within every ε-neighborhood of
L.See Figure 6.1.A sequence that is not convergent is said to be divergent.
The sequence {1+i n } illustrated in (1) is divergent since the general term
zn =1+i n does not approach a fixed complex number as n →∞.Indeed,
you should verify that the first four terms of this sequence repeat endlessly as
n increases.
EXAMPLE 1 AConvergent Sequence
� � n+1 i
The sequence
n
i
converges since lim
n→∞
n+1
= 0.As we see from
n
−1, − i
2
, 1
3
, i
4
, −1 , ··· ,
5
and Figure 6.2, the terms of the sequence, marked by colored dots in the
figure, spiral in toward the point z =0asn increases.
The following theorem for sequences is the analogue of Theorem 2.1 in
Section 2.6.
(1)