Graphics Reference
In-Depth Information
(c) Find an integer
k
, such that if
n
k
, then
a
a
.
(d) By considering that
a
a
and that
a
a
, etc.,
provethat
a
(
)
a
.
(e) Why is ((
) a null sequence? You have chosen
k
in part (c)
and it remains fixed.
(f) Use the squeeze rule (qn 36(d)) to show that (
n
/2
) is a null
sequence.
(g) What numbers might have been used in place of
)
a
in part (c)
(with perhaps a different
k
) which would still haveled to a
proof that the sequence was null?
41 Use the method of qn 40 to establish that the following sequences
arenull:
/
n
!),
(iii) (
n
!/
n
), using qn 2.45,(iv) (
n
(0.9)
).
(i) (
n
/2
),
(ii) (2
42 Use the absolute value rule (qn 33) to extend qn 39 and establish
that (
c
ยท
x
) is a null sequence when
1
x
1.
43
The product rule for null sequences
Let both (
a
) and (
b
) be null sequences, and suppose
0 is given.
(i) Must there be an
N
such that
a
when
n
N
?
1 when
n
N
?
(iii) Can you find an
N
such that when
n
(ii) Must there be an
N
such that
b
N
, then both
n
N
and
n
N
?
(iv) If
n
N
, must
a
?
You have proved that the termwise product (
a
b
b
) of two null
sequences is null.
(v) If the sequence (
c
) is also null, what about (
a
b
c
)? And the
termwise product of
k
null sequences?
44
The sum rule for null sequences
Let both (
a
) and (
b
) be null sequences, and suppose
0 is given.
(i) Must there be an
N
such that
a
/2 when
n
N
?
(ii) Must there be an
N
b
/2 when
n
N
?
(iii) Is there an
N
such that when
n
N
, then both
n
N
and
n
N
such that
?
(iv) If
n
N
, must
a
b
? Usethetriangleinequality (qn
2.61).
You have proved that the termwise sum (
a
b
) of two null
sequences is null.
