infinity 269

1

–

2

π

–

2

which approaches the value as n→∞. Thus we can

write .

By taking the

LOGARITHM

, we see that an infinite

product with positive terms converges if, and

only if, the

SERIES

converges. (This is why

mathematicians do not deem convergence to zero as a

valid form of convergence for an infinite product: the

value of the sum would then be –∞) Mathe-

maticians have shown that a series converges

absolutely exactly if converges.

In 1656 English mathematician J

OHN

W

ALLIS

dis-

covered the following remarkable representation for

as an infinite product:

This expression is today called W

ALLIS

’

S PRODUCT

.

See also

ABSOLUTE CONVERGENCE

;

ZETA FUNCTION

.

infinitesimal A positive quantity, supposedly infin-

itely small yet not itself zero, is called an infinitesimal.

The abstract existence of such quantities played an

important, yet troublesome, role in the early develop-

ment of

CALCULUS

. The theoretical difficulties incurred

by them were later made moot when A

UGUSTIN

-L

OUIS

C

AUCHY

(1789–1857) introduced the notion of a

LIMIT

.

Cauchy’s work put calculus on a sound theoretical set-

ting and removed the need to ever speak in terms of

infinitesimals. The notion is now considered obsolete.

(Although mathematicians have recently developed a

theory of nonstandard analysis that does, in some sense,

include a valid concept of an infinitesimal, after all.)

See also

FLUXION

;

HISTORY OF CALCULUS

(essay).

infinity In common usage, the word infinite is used to

denote something that is unbounded, limitless, and end-

less. For example, the set of counting numbers 1, 2, 3, …

is unbounded (after any given counting number there is

always another), and the set of these numbers is said to

be infinite. In geometry, a straight line is usually per-

ceived as without end, and so it is infinite in extent. In

the physical world, however, there is no clear example of

infinite quantity (or, if there were, we as humans would

never be able to fully perceive it). Physicists have come to

believe, for instance, that there are only a finite number

of atoms in the universe (on the order of 1087 atoms) and

that the universe is closed in shape and not of infinite

extent. If this is indeed the case, then it would never be

possible to draw an infinite line (there are not enough

atoms for the ink), and if the universe is indeed bounded,

the line may eventually loop back on itself. Nonetheless,

the infinite is an abstract concept we feel we can, on

some level, comprehend. It does, however, bring with it

many paradoxical difficulties, as demonstrated by

Z

ENO

’

S PARADOXES

, H

ILBERT

’

S INFINITE HOTEL

, and the

T

RISTRAM

S

HANDY PARADOX

, for example.

The notion of something that is unlimited arises in

mathematics in a number of varied settings. For instance:

1. Limits: In 1655 English mathematician John

Wallis introduced the symbol ∞as shorthand for the

phrase “becoming large and more positive.” This nota-

tion is used today, in particular, in the study of

LIMIT

s.

For example, the statement limn→∞ = 5 is

to be read: “the quantity approaches the

value 5 as nbecomes large and positive.” The equation

limn→∞ = –∞reads: “the function becomes

large and negative as xbecomes large and positive.”

This means that for any negative number –Mone cares