Complex Analysis - Maths KU

306 Chapter 6 Series and Residues
Definition 6.1 Absolute and Conditional Convergence
An infinite series � ∞
k=1 zk is said to be absolutely convergent if
� ∞
k=1 |zk| converges.An infinite series � ∞
k=1 zk is said to be conditionally
convergent if it converges but � ∞
k=1 |zk| diverges.
In elementary calculus a real series of the form ∞� 1
is called a p-series
k=1 kp and converges for p>1 and diverges for p ≤ 1.We use this well-known result
in the next example.
EXAMPLE 4 Absolute Convergence
The series ∞� i
k=1
k
∞�
�
�
is absolutely convergent since the series �
i
k2 �
k=1
k
k2 �
�
�
� is the same
as the real convergent p-series ∞� 1
.Here we identify p =2> 1.
k2 As in real calculus:
k=1
Absolute convergence implies convergence.
See Problem 47 in Exercises 6.1. We are able to conclude that the series in
Example 4,
∞�
k=1
ik 1 i
= i − − + ···
k2 22 32 converges because it is was shown to be absolutely convergent.
Tests for Convergence Two of the most frequently used tests for
convergence of infinite series are given in the next theorems.
Theorem 6.4 Ratio Test
Suppose �∞ k=1 zk is a series of nonzero complex terms such that
lim
� �
�zn+1
�
� �
� � = L. (9)
n→∞
(i) IfL1orL = ∞, then the series diverges.
zn
(iii) IfL = 1, the test is inconclusive.