series 459

=

1

–

32

1

–

16

1

–

8

1

–

4

1

–

2

…

numbers. (For instance, √

–

2 is the limit of the sequence 1,

1.4, 1.41, 1.414, 1.4142,…) As such, it is possible to

define a quantity raised to an irrational power as a limit

of rational powers, which are well defined. (For example,

3√

–

2is the limit of the sequence

.) Any infinite sum that is a

SERIES

can be

thought of as a

LIMIT

of a sequence of

PARTIAL SUM

s,

and any

INFINITE PRODUCT

can be considered to be the

limit of a sequence of partial products.

See also

ARITHMETIC SEQUENCE

;

BOUND

;

GEOMET

-

RIC SEQUENCE

;

HARMONIC SEQUENCE

.

series A sum of numbers is called a series. The sum

could be finite, such as 2 + 4 + 6 + 8 + 10, for example,

or it could be an infinite sum, as for 2 + 5 + 8 + 11 +

14 + … for instance. Each number in the sum is called

a term of the series.

In 1755 L

EONHARD

E

ULER

introduced use of the

Greek letter sigma Σ(S for “sum”) to abbreviate the

writing of a series. For example, the sum of the first

five even numbers can be written and the sum of

all numbers 1 less than a multiple of three as .

(See

SUMMATION

.)

There are no conceptual difficulties with the notion

of summing just a finite collection of numbers. How-

ever, understanding what we mean by an infinite sum is

a delicate matter—such sums can exhibit very different

characters. Some series clearly do not add to a finite

value (as for 2 + 5 + 8 + 11 + 14, …, for example),

whereas one could argue that other infinite series do.

(The act of walking from one side of the room to the

other, for instance, suggests that the series + +

+ + +… “adds” to the value 1: first walk

halfway across the room, and then half the remaining

distance, and then the half the distance that remains,

and so on.)

In some cases the situation is not at all clear. For

example, the series 1 – 1 + 1 – 1 + 1 – 1 + … of positive

and negative terms seems to oscillate: an even number

of terms add to zero, an odd number to 1, and it does

not seem possible to assign a single sum to this series.

(Some mathematicians in the past argued that the sum

of this series should be 1/2, a value between zero and 1.)

These different behaviors of infinite series caused

scholars much confusion over the centuries. Matters

were not properly resolved until A

UGUSTIN

-L

OUIS

C

AUCHY

(1789–1857) introduced the notion of a

LIMIT

, and was able to apply it to the study of infinite

sums. Today we say that an infinite series = a1+

a2+ a3+… converges to a finite value Lif the

SEQUENCE

of

PARTIAL SUMS

:

S1= a1

S2= a1+ a2

S3= a1+ a2+ a3

tends to Lin the limit as n→∞. For example, the infi-

nite series

has partial sums:

that approach the value 1. In this sense we say that the

sum equals 1. (In a practical

sense, however, adding the terms of the series only

brings us closer and closer to the value 1: we can never

attain it. We are physically limited by the fact that we

cannot keep adding terms indefinitely.)

An infinite series that converges is called a conver-

gent series. Any series that does not converge to a finite

value (either because the value of the sum seems to be

infinite, or because the partial sums oscillate) is called

divergent. There are a number of tests to determine

whether or not a series converges or diverges. (See

CONVERGENT SERIES

.)