Graphics Reference
In-Depth Information
a
a
(a)
a
.
(b)
a
.
(c)
a
a
.
(d)
a
a
.
(i)
a
n
,
(ii)
a
1/
n
,
(iii)
a
(
1)
,
(iv)
a
n
,
a
n
,
(v)
a
1/
n
,
(vi)
a
1,
2
.
When property (a) in qn 4 holds, the sequence is said to be
strictly
monotonic increasing
or just
strictly increasing
.
When property (b) holds, the sequence is said to be
monotonic
increasing
or just
increasing
.
When property (c) holds, the sequence is said to be
strictly
monotonic decreasing
or
strictly decreasing
.
When property (d) holds, the sequence is said to be
monotonic
decreasing
or
decreasing
.
When any one of (a), (b), (c) or (d) holds, the sequence is said to be
monotonic
.
The seemingly bizarre status of sequence 4(vi) is a consequence of
wanting useful definitions. The fact that constant sequences satisfy
conditions (b) and (d) simultaneously is not a suMcient reason to revise
our definitions, because experience has shown that these definitions, as
they stand, lead to simple statements of theorems and proofs.
(vii)
a
Bounded sequences
5 Test each of the sequences (
a
) (i)
—
(iv) defined here, to determine
whether either or both of the following properties applies. Sketch a
graph of the first few terms.
(a) There is a number
U
such that
a
U
for all
n
.
(b) There is a number
L
such that
L
a
for all
n
.
(i)
a
(
2)
,
(ii)
a
(
1)
/
n
,
n
.
When property (a) in qn 5 holds, the sequence (
a
(iii)
a
sin
n
,
(iv)
a
) is said to be
bounded above
, and thenumbr
U
is called an
upper bound
of the
sequence.
When property (b) holds, the sequence is said to be
bounded below
,
and thenumbr
L
is called a
lower bound
of the sequence.
When both properties (a) and (b) hold, so that there are numbers
L
and
U
such that
L
a
U
, for all values of
n
, the sequence (
a
) is said
to be
bounded
.
A sequence which is neither bounded above nor bounded below is
