Graphics Reference
In-Depth Information
a
n
ε
0
n
N
or
N
or
N
ε
Fig 3.28
30 Prove that the sequence (1/
n
) is a null sequence.
31 If (
a
)
, provethat (1/
a
) is a null sequence.
If (
a
) is a null sequence and
a
0 for all
n
, provethat
(1/
a
)
.
Give an example to show that it is possible to have a null sequence
(
a
) without having (1/
a
)
.
32
The scalar rule for null sequences
Let (
a
) be a null sequence and
c
a constant number.
Provethat (
c
·
a
) is a null sequence.
Consider the cases
c
0 and
c
0, in turn.
Deduce that (10/
n
) is a null sequence.
33
The absolute value rule for null sequences
(a) Let (
a
) be a null sequence. Prove that (
a
) is a null
sequence.
(b) Conversely, let (
a
) be a null sequence. Prove that (
a
)isa
null sequence.
34
A squeeze rule or sandwich theorem for null sequences
I
Let (
a
) be a null sequence, and 0
b
a
, for all
n
. Provethat
(
b
) is a null sequence.
35 Prove that each of the following sequences is null:
(i) (1/(
n
1)),
(ii) (10/(
n
1)),
(iii) (20/(7
n
3)),
(iv) (1/(
n
1)),
(v) (
(
n
1)
n
),
(vi) ((sin
n
)/
n
),
(vii) (sin
n
),
(viii) (sin(
n
!
·
c
)), where
c
is a rational number,
(ix) (
a
) where
a
1/(2
n
1) and
a
1/
n
.
Standard properties of the sine function should be used: they will
be proved in chapter 11.
