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312 limit

rates for illnesses in different age groups, or popula-

tions with different habits. Life tables are regularly

updated to take account of new factors that may alter

life expectancy.

In his 1662 pamphlet, Natural and Political Obser-

vations Made upon the Bills of Mortality, English shop-

keeper John Graunt was the first to collate and publish

tables of mortality for a specific population. In summa-

rizing government burial records for the years 1604–61,

Graunt was able to estimate the number of deaths,

decade by decade, to expect among a group of typical

100 Londoners born at the same time. He gave the

name “life table” to his display of results. Graunt was

also able to make general observations about the popu-

lation—that women live longer than men, that the

death rate is typically constant, and the like—and he

was the first to comment on the regularity of social phe-

nomena in this way.

In 1693, relying on records collated by Casper

Heumann of Breslau, English astronomer Edmund

Halley refined the mathematical techniques used by

Graunt to compile a revised and more detailed set of

mortality tables, ones suitable for properly analyzing

annuities.

A

CTUARIAL SCIENCE

is the mathematical study life

expectancies and other demographic trends. Halley’s

work is said to be the founding work in this field.

See also

HISTORY OF PROBABILITY AND STATISTICS

(essay);

STATISTICS

.

limit Intuitively, a limit is a quantity that can be

approached more and more closely but not necessarily

ever reached. For example, the numbers in the

SE

-

QUENCE

0.9, 0.99, 0.999,… approach, but never reach,

the value 1. We say that the limit of this se-

quence is one. The function f(x) = takes values

closer and closer to zero as xbecomes large. We say the

limit of this function as xbecomes large is zero.

The notion of a limit was first properly identified

by the French mathematician A

UGUSTIN

-L

OUIS

C

AUCHY

(1789–1857), arising as a necessary tool for placing the

theory of calculus on sound theoretical footing. Ger-

man mathematician K

ARL

T

HEODOR

W

ILHELM

W

EIER

-

STRASS

(1815–97) later developed the idea further and

gave the concept of a limit the precise, rigorous defini-

tions we follow today.

The Limit of a Sequence

A sequence of numbers a1,a2,a3,… has limit L

if one can demonstrate that for any positive

number ε(no matter how small), eventually

all the numbers in the sequence will be this

close to the value L. That is, one can find a

value Nso that anlies between L– εand L+ ε

if n> N.

This says that no matter which level of precision

you care to choose (ε), eventually all the numbers in the

list (from aNonward) will be within a distance εfrom

L. For example, in the sequence 0.9, 0.99, 0.999,…, all

the numbers in the list from the third place onward are

within a distance 1/1000 from the value 1, and all

numbers from the sixth place onward are within one-

millionth of the value 1. In fact, for any small value ε,

we can locate a position in the sequence so that from

that position onward, all values in the list are within a

distance εfrom one.

If a sequence {an} has a limit value L, we write:

limn→∞an= L. For example, one can show that limn→∞

= 0 and limn→∞ . A careful study of

CONVER

-

GENT SEQUENCE

s shows that not all sequences, however,

have a limit.

Any infinite sum (

SERIES

) can be thought of as a

limit of a sequence of

PARTIAL SUM

s, and any

INFINITE

PRODUCT

the limit of a sequence of partial products.

The Limit of a Function

A function f(x) has limit Las xbecomes large

if one can demonstrate that for any positive

number ε(no matter how small), all the out-

puts of the function will eventually be this

close to the value L. That is, one can find a

number Nso that the value f(x) lies between

L– εand L+ εif x> N.

This says that no matter which level of precision

you care to choose (ε), eventually all outputs of the

function f(x) (from x= Nonward) will be within a dis-

tance εfrom L. For example, all outputs of the function

f(x) = are smaller than 0.001 from the point x= 1,000

onward. Similarly, all outputs are smaller than 0.000001

1

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