Graphics Reference
In-Depth Information
b
c
(v) If the sequence (
c
) is also null, what about (
a
)? And
thetrmwisesum of
k
null sequences?
45 Let both (
a
) and (
b
) be null sequences. Show that (
c
·
a
d
·
b
)is
a null sequence, where
c
and
d
areconstants. Usethescalar rule
and thesum rul.
The difference rule
By choosing suitablevalus for
c
and
d
, show that (
a
b
) is a null
sequence.
46
(i)
A squeeze rule or sandwich theorem for null sequences
II
Let both (
a
) and (
c
) be null sequences, and
a
b
c
, for
all
n
.
(a) Say why (
c
a
) is a null sequence.
(b) Say why (
b
) is a null sequence.
(c) Finally say why (
b
a
) is a null sequence.
(ii)
The squeeze rule with a shift
II
Let both (
a
) and (
c
) be null sequences, and
a
b
c
, for
n
k
. Say why (
b
) must be a null sequence.
47 Let
A
.
Give an example of a sequence (
a
a
a
...
a
) which is not null, but for which
the sequence (
A
/
n
) is null.
Summary
-
null sequences
Definition
A sequence (
a
) is said to tend to zero or to be a
null sequence, if and only if, given any
0,
there exists an
N
such that
n
N
a
.
This is expressed symbolically by writing
(
a
)
0as
n
.
Theorem
(1/
n
) is a null sequence.
qn 29
Theorem
(
x
) is a null sequence, when
1
x
1.
qns
39, 42
The scalar rule for null sequences
qn 32
If (
a
) is a null sequence, then (
c
·
a
) is a null
sequence.
The absolute value rule for null sequences
qn 33
(
a
) is a null sequence
(
a
) is a null
sequence.
