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(ii) Let
y
arctan
x
,so
dy
1
1
x
dx
.
Provethat for
n
1,
(1
0.
Deduce that when
x
0,
y
0 and
y
(
1)
(2
n
)!
x
)
y
2(
n
1)
xy
n
(
n
1)
y
There is ambiguity in the literature on the question of whether N
contains 0 or not. If mathematics starts with sets, and particularly the
empty set, the equipment for counting the elements of finite sets must
include 0. The psychological origins of counting, however, start with 1.
It is this convention we have called
natural
. Both conventions are
well-established.
A proposition,
P
(
n
), relating to a natural number
n
, is valid for all
natural numbers
n
,if
(a)
P
(1) is true; the proposition is valid for
n
1 and
(b)
P
(
n
)
P
(
n
1); theproposition for
n
implies the proposition for
n
1.
Historical note
Triangular numbers were studied by Nicomachus (c. 100 A.D.) The
sum of thefirst
n
squares was known to Archimedes (c. 250 B.C.) and
thesum of thefirst
n
cubes to the Arabs (c. A.D. 1010). The justification
of these forms uses induction implicitly.
The earliest use of proof by mathematical induction in the literature
is by Maurolycus in his study of polygonal numbers published in
Venice in 1575. Pascal knew the method of Marolycus and used it for
work on the binomial coeMcients (c. 1657). In 1713, Jacques Bernoulli
used an inductive proof to make rigorous the claim of Wallis that
r
n
1
k
1
as
n
.
By the latter part of the eighteenth century, induction was being used
by several authors. The name 'mathematical induction' as distinct from
scientific induction is due to De Morgan (1838). The use of inductive
definitions long pre-dates this method of proof.
The well-ordering principle, that every non-empty set of natural
numbers has a least member, is equivalent to mathematical induction
(see Ledermann and Weir, 1996). Well-ordering was used by Euclid
