rates for illnesses in different age groups, or populations 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 Observations Made upon the Bills of Mortality, English shopkeeper John Graunt was the first to collate and publish
tables of mortality for a specific population. In summarizing 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 population—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 phenomena 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.
ACTUARIAL 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 SEQUENCE 0.9, 0.99, 0.999,… approach, but never reach,
the value 1. We say that the limit of this sequence is one. The function f (x) =
takes values
closer and closer to zero as x becomes large. We say the
limit of this function as x becomes large is zero.
The notion of a limit was first properly identified
by the French mathematician AUGUSTIN-LOUIS CAUCHY
(1789–1857), arising as a necessary tool for placing the
theory of calculus on sound theoretical footing. German mathematician KARL THEODOR WILHELM WEIERSTRASS (1815–97) later developed the idea further and
gave the concept of a limit the precise, rigorous definitions we follow today.
The Limit of a Sequence
A sequence of numbers a 1 ,a 2 ,a 3 ,… 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 N so that a n lies between L – ε and 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 a N onward) will be within a distance ε from
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 onemillionth 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 ε from one.
If a sequence {a n } has a limit value L, we write:
lim n→∞ a n = L. For example, one can show that lim n→∞
= 0 and lim n→∞
. A careful study of CONVERGENT SEQUENCEs 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 SUMs, and any INFINITE
PRODUCT the limit of a sequence of partial products.
The Limit of a Function
A function f(x) has limit L as x becomes large
if one can demonstrate that for any positive
number ε (no matter how small), all the outputs of the function will eventually be this
close to the value L. That is, one can find a
number N so that the value f(x) lies between
L – ε and L + ε if 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 = N onward) will be within a distance ε from 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
– x
n
n
2
2
1 1
− =
1
– n
1
– x
312 limit
updated to take account of new factors that may alter
life expectancy.
In his 1662 pamphlet, Natural and Political Observations Made upon the Bills of Mortality, English shopkeeper John Graunt was the first to collate and publish
tables of mortality for a specific population. In summarizing 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 population—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 phenomena 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.
ACTUARIAL 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 SEQUENCE 0.9, 0.99, 0.999,… approach, but never reach,
the value 1. We say that the limit of this sequence is one. The function f (x) =
takes values
closer and closer to zero as x becomes large. We say the
limit of this function as x becomes large is zero.
The notion of a limit was first properly identified
by the French mathematician AUGUSTIN-LOUIS CAUCHY
(1789–1857), arising as a necessary tool for placing the
theory of calculus on sound theoretical footing. German mathematician KARL THEODOR WILHELM WEIERSTRASS (1815–97) later developed the idea further and
gave the concept of a limit the precise, rigorous definitions we follow today.
The Limit of a Sequence
A sequence of numbers a 1 ,a 2 ,a 3 ,… 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 N so that a n lies between L – ε and 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 a N onward) will be within a distance ε from
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 onemillionth 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 ε from one.
If a sequence {a n } has a limit value L, we write:
lim n→∞ a n = L. For example, one can show that lim n→∞
= 0 and lim n→∞
. A careful study of CONVERGENT SEQUENCEs 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 SUMs, and any INFINITE
PRODUCT the limit of a sequence of partial products.
The Limit of a Function
A function f(x) has limit L as x becomes large
if one can demonstrate that for any positive
number ε (no matter how small), all the outputs of the function will eventually be this
close to the value L. That is, one can find a
number N so that the value f(x) lies between
L – ε and L + ε if 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 = N onward) will be within a distance ε from 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
– x
n
n
2
2
1 1
− =
1
– n
1
– x
312 limit