4 absolute value

Let pn= |an|– an. Then each value pnis either

zero or equal to 2 |an|, depending on whether

anis positive or negative. In particular we have

that 0 ≤pn≤2|an|.

Consequently, and so, by

the

COMPARISON TEST

, converges. Conse-

quently

so does .

This test does not cover all cases, however. It is still

possible that a series containing positive and

negative terms might converge even though

does not. For example, the alternating

HARMONIC

SERIES

converges, yet

does not. A series that

converges “on the condition that the negative signs

remain present,” that is, one for which

converges but does not, is called “condition-

ally convergent.” Manipulating conditionally convergent

series can lead to all sorts of paradoxes. For example,

the following argument “proves” that 1 = 2:

Start with the observation that:

(This follows from the study of the harmonic

series or from M

ERCATOR

’

S EXPANSION

.) Con-

sequently:

Collecting terms with a common denominator

gives:

and so 2 = 1.

Paradoxes like these show that it is not permissible

to rearrange the order of terms of a conditionally con-

vergent series. Mathematicians have shown, however,

that rearranging the terms of an absolutely convergent

series is valid.

See also

ABSOLUTE VALUE

.

absolute value (modulus) Loosely speaking, the

absolute value of a

REAL NUMBER

is the “positive ver-

sion of that number.” Vertical bars are used to denote

the absolute value of a number. For example, the abso-

lute value of negative three is |–3|= 3, and the absolute

value of four is |4|= 4. The absolute value of a real

number ais typically envisioned three ways:

1. |a|equals aitself if ais positive or zero, and equals

–aif it is negative. (For example, |–3|= –(–3) = 3

and |3|= 3.)

2. |a|equals the positive square root of a2. (For example,

.)

3. |a|is the distance between the points aand 0 on

the real number line. (For example, |–3|= 3 = |3|

since both –3 and 3 are three units from the ori-

gin.) More generally, if aand bare two points on

the number line, then the distance between them

on the number line is given by |a– b|. (For exam-

ple, the points 4 and –7 are |4–(–7)|= |4 + 7|= 11

units apart.)

By examining each of the cases with aand bposi-

tive or negative, one can check that the absolute value

function satisfies the following properties:

i. |a+ b|≤|a|+ |b|

ii. |a– b|≤|a|+ |b|

iii. |a · b|= |a|· |b|

Knowing the absolute value of a quantity deter-

mines the value of that quantity up to sign. For exam-

ple, the equation |x+2|= 5 tells us that either x+ 2 = 5