12 1. The Complex Numbers

∑∞

k=0

zk are convergent and, by Theorem 1.2.4, so is the sequence {sn} itself. Thus,

the series converges.

It follows from the triangle inequality that

|sn| =

n

k=0

zk ≤

n

k=0

|zk|.

The inequality of the theorem follows when we pass to the limit as n → ∞ in this

inequality and use the result of Exercise 1.2.7.

Example 1.2.9. Prove that the complex series

∑∞

k=1

(k + k2i)−1 converges abso-

lutely.

Solution: By the second form of the triangle inequality (see Exercise 1.2.2)

we have

|k +

k2i|

≥

k2

− k ≥

k2

2

if k ≥ 2.

Thus,

|k +

k2i|−1

≤

2k−2

if k ≥ 2.

Since the p-series

∑∞

k=1

k−2 converges, so does

∑∞

k=1

2k−2 and, by comparison,

∑∞

k=1

|k +

k2i|−1.

Thus,

∑∞

k=1

(k +

k2i)−1

converges absolutely.

Power Series. A complex power series is a series of the form

(1.2.2)

∞

n=0

an(z −

z0)n,

where the coeﬃcients an are complex numbers, z0 is a complex number and z is a

complex variable. A power series of this form is said to be centered at z0. It defines

a complex function of the complex variable z, with domain the set of those z ∈ C

for which the series converges.

Remark 1.2.10. We know from calculus that the set on which a real power series

∞

n=0

an(x −

x0)n

converges is an interval – the interval of convergence – consisting of an open interval

centered at x0 and possibly one or both of its endpoints. The power series converges

absolutely at each point of the open interval. The radius of this interval is called

the radius of convergence of the power series and is computed using the root test

or the ratio test. As we shall see in Chapter 3, a similar result is true for complex

power series, but the interval of convergence is replaced by a disc of convergence.

If for r 0 we set

Dr(z0) = {z ∈ C : |z| r} and Dr(z0) = {z ∈ C : |z| ≤ r},

then Dr(z0) is called the open disc of radius r, centered at z0, while Dr(z0) is

called the closed disc of radius r, centered at z0. Given a power series, centered at

z0, there is a number R ≥ 0, called the radius of convergence of the power series,

such that the series converges absolutely for each z ∈ DR(z0) and diverges for each

z / ∈ DR(z0). When we study power series in detail, we will prove this result and tell

how to calculate R in general. However, for most of the series we will be studying,