1549901369-Elements_of_Real_Analysis__Denlinger_

480 Chapter 8 • Infinite Series of Real Numbers

CAUTION: In using the alternating series test, students often ask whether

it is necessary to include the hypothesis that {an} is monotone decreasing. In-

deed, students will often disregard this hypothesis when deciding whether the

alternating se ries test is applicable to a particular series. The following exam-

ple is offered to show that this test cannot be used when {an} is not monotone

decreasing.

Example 8.3.6 The alternating series

1-2 +1- 2 1 +1-2 3 344 +1-2+1-2 5 5 + ... +.!.-nn 2+ ...

diverges, even though an----+ 0.

Proof. Exercise 1. 0

ABSOLUTE CONVERGENCE

Definition 8.3. 7 A series Lan is said to converge absolutely (or, be ab-

solutely convergent) if L lanl converges. A series that converges, but not

absolutely, is said to converge conditionally.^3

For example, the alternating harmonic series L (-l~n+i converges condi-
tionally, as seen in Examples 2.5.16 and 8.3.3.
As we shall see, absolute convergence is stronger t han convergence. We
shall show in what follows that an absolutely convergent series must converge,
but the converse is not true. To gain an understanding of this concept, and
closely related facts, it will be helpful to define two series intimately related to
a given series.

Definition 8.3.8 Given a sequence {an} of real numbers, we define the se-
quences { a;t} and {a;;} by

a;t' =max{ an, O} and a;; = max{-an, O}.

Lemma 8.3.9 For any a E JR,
(a) a+ 2': 0 and a-2': O;
(b) a=a+-a-;
(c) lal =a++ a-.

Proof. Exercise 2. •

3. The term "conditionally" convergent w ill b e explained more fully later in t his section.