15

16

7

8

3

4

1

2

102 convergent improper integral

2

1

5

1

4

1

3

1

2

6

7

5

6

4

5

3

4

2

3

1

2

+

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 TABLE

s.)

In mathematics it is sometimes convenient to prove

the contrapositive form of a theorem rather than prove

the assertion directly. This approach is called contra-

positive reasoning (or modus tollens), and the proof

presented is a proof by contraposition. For example,

the theorem: if n2is odd, then n is odd, is best proved

by assuming that nis even (that is, n= 2kfor some

integer k) and then showing that n2is also even.

The contrapositive of a conditional “p implies q”

should not be confused with the inverse of the state-

ment: “not p implies not q.” This variation is not a log-

ically equivalent form of the original conditional.

See also

ARGUMENT

;

CONVERSE

;

PROOF

.

convergent improper integral See

IMPROPER

INTEGRAL

.

convergent sequence A

SEQUENCE

of numbers

a1,a2,a3,… is said to converge if the terms of the

sequence become arbitrarily close to, but do not nec-

essarily 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 {an} converges to limit L, we

write limn→∞an= L, or, alternatively, an→Las n→∞,

which is read as “anapproaches Las nbecomes 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 mathe-

matically precise with an “ε–Ndefinition” 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 exam-

ple), 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 Lif the sequence

of

PARTIAL SUMS

, Sn= a1+ a2+…+an, approaches the

value Lin the

LIMIT

as n→∞. To illustrate, the series

has partial sums:

which approach the value 1 as ngrows. 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: