Graphics Reference
In-Depth Information
C
-1
0
1
2
[
C
]
[
C +
1]
19
(i) Rewrite [
A
]
A
[
A
]
1 in the equivalent form
...
[
A
]
....
(ii) (a) For each real number
x
, could there be an integer
N
(
x
)
such that
N
(
x
)
x
N
(
x
)
1?
(b) For each real number
x
, could there be an integer
M
(
x
)
such that
M
(
x
)
x
M
(
x
1)?
(c) For each real number
x
, you know how to find an integer
K
(
x
) such that
K
(
x
)
x
K
(
x
1). Can you find more
than oneway of defining
K
(
x
) to satisfy this condition?
20 Provethat (
n
)
as
n
.
21
(a) Let
y
be a fixed positive number. Show that the sequence
(
ny
)
.
(b) Let
x
1, so that for somepositive
y
,
x
1
y
.Use
Bernoulli's inequality (qn 2.29), to show that
x
1
ny
, and
deduce that (
x
)
.
22 If a sequence tends to
, must it be(monotonic) incrasing?
23 Construct a definition for '(
a
'.
Check that your definition guarantees that the sequence defined by
a
)
as
n
n
tends to
.
Summary
-
the language of sequences
The symbol for a sequence, (
a
), denotes the infinite list of numbers
a
,
a
,
a
, ...,
a
,...
in this order.
n
N.
Definition
qn 4
A sequence (
a
) is said to be
increasing
when
a
a
for all
n
, and is said to be
decreasing
when
a
for all
n
. Both increasing and
decreasing sequences are called
monotonic
.
a
Definition
qn 5
A sequence (
a
) is said to be
bounded above
when there is a number
U
such that
a
U
for
all
n
.
A sequence (
a
) is said to be
bounded below
when there is a number
L
such that
L
a
for
all
n
.
