logically equivalent form of the statement, and so can
be used at any time in its stead. (One shows that p → q
and (¬q) → (¬p) have identical TRUTH TABLEs.)
In mathematics it is sometimes convenient to prove
the contrapositive form of a theorem rather than prove
the assertion directly. This approach is called contrapositive reasoning (or modus tollens), and the proof
presented is a proof by contraposition. For example,
the theorem: if n
2 is odd, then n is odd, is best proved
by assuming that n is even (that is, n = 2k for some
integer k) and then showing that n
2 is also even.
The contrapositive of a conditional “p implies q”
should not be confused with the inverse of the statement: “not p implies not q.” This variation is not a logically equivalent form of the original conditional.
See also ARGUMENT; CONVERSE; PROOF.
convergent improper integral See IMPROPER
INTEGRAL.
convergent sequence A SEQUENCE of numbers
a 1 ,a 2 ,a 3 ,… is said to converge if the terms of the
sequence become arbitrarily close to, but do not necessarily ever reach, a particular finite value L. For
example, the numbers in the sequence 0.9, 0.99,
0.999,… approach the value 1. We call 1 the LIMIT of
this sequence.
Any sequence that converges is called a convergent
sequence. If a sequence {a n } converges to limit L, we
write lim n→∞ a n = L, or, alternatively, a n → L as n → ∞,
which is read as “a n approaches L as n becomes large.”
For example, the sequence , , , ,… has limit one
(
), and the sequence 1,– , ,– , ,…
has limit zero (
as n → ∞). The
notions of limit and convergence can be made mathematically precise with an “ε –N definition” of a limit.
(See LIMIT.)
A sequence that does not converge is said to
diverge. A divergent sequence could have terms that
grow in size without bound (1,4,9,16,25,…, for example), terms that oscillate without converging to a limit
( ,– , ,– , ,– ,…, for example), or terms that
oscillate without bound (1,2,1,3,1,4,1,5,1,6,1,7,1,…,
for instance).
See also DIVERGENT; INFINITE PRODUCT; SERIES.
convergent series An infinite SERIES
is said to converge to a value L if the sequence
of PARTIAL SUMS, S n = a 1 + a 2 +…+a n , approaches the
value L in the LIMIT as n→∞. To illustrate, the series
has partial sums:
which approach the value 1 as n grows. In this sense we
say that the series
converges to 1, and we write:
If the limit of the partial sums does not exist, then
the series is said to diverge. For example, the series
1 – 1 + 1 – 1 + 1 – … diverges because the partial sums
oscillate between being 1 and 0 and never settle to a
particular value. The series 1 + 2 + 3 + 4 + … diverges
because the partial sums grow arbitrarily large. The
series
diverges for the same reason, which can
be seen as follows:
S
n
n
n
n
n
n = +
+
+ +
>
+
+
+ +
1
1
1
2
1
3
1
1
1
1
1
L
L
1
1 n
n=
∞
∑
1
2
1
4
1
8
1
16
1
+ + +
+ =
L
1
2
1
n
n=
∞
∑
S
S
S
S n
n
1
2
3
1
2
1
2
1
4
3
4
1
2
1
4
1
8
7
8
1
1
2
=
= + =
= + + =
= −
M
M
1
2
1
2
1
4
1
8
1
16
1
n
n
= + + +
+
=
∞
∑
L
a + …
3
a a a
n
n
= + +
=
∞
∑
1
1
2
6
7
5
6
4
5
3
4
2
3
1
2
lim
( )
n
n
n
→∞
+
−
→
1
0
1
1
5
1
4
1
3
1
2
lim n
n
n
→∞
− =
2 1
2
1
15
16
7
8
3
4
1
2
102 convergent improper integral