Graphics Reference
In-Depth Information
1
1
(v)
n
th term
n
. Useqn 30, thescalar rule
(qn 32) and the squeeze rule (qn 34).
n
n
1
2
1
n
. Use qn 29, the squeeze rule (qn 34) and the
absolutevaluerule(qn 33).
(vii) All terms are 0.
(viii) If
c
p
/
q
, where
p
and
q
are integers,
q
positive, then when
n
q
, thetrms are0.
(ix) 0
a
(vi)
sin
n
n
2/
n
. Useqn 29, thescalar rule(qn 32) and the
squeeze rule (qn 34).
36
(a) Yes. If
n
N
a
, then
n
N
a
.
(b) No. See qn 26(v).
(c) [
n
N
a
]
[
n
N
k
a
].
(d) (
a
) null
(
a
) null, by (a),
b
a
for
n
k
b
a
0
0
for all
n
.
So (
b
) is a null sequence by the squeeze rule (qn 34), and
(
b
) is a null sequence by (c).
37
(i) and (ii) are subsequences of (1/
n
). Useqn 36(a), the
subsequence rule.
(iii) for
n
10, terms
1/
n
, use36(d) with
k
10.
(iv) is a subsequence of (1/
n
) for
n
34. Use36(a) to claim that
the subsequence is null and the shift rule (qn 36(c)) with
k
34.
38
(i)
c
for every positive
,so
c
0 and
c
0.
(ii) A constant sequence with each term
a
b
would haveto
be a null sequence by the squeeze rule, and therefore
a
b
from part (i).
39
(d) (1/
n
) is a null sequence by qn 29, so (1/
yn
) is a null sequence
by thescalar rule(qn 32), so (1/(
yn
1)) is a null sequence by
the squeeze rule (qn 34) and finally (
x
) is a null sequence by
the squeeze rule (qn 34).
(e) Use the scalar rule (qn 32).
40
(c)
k
5.
(d) For example,
a
(
)
a
.
(e) From qn 39(e).
(f) From (d) and the squeeze rule (qn 34) (
a
) is a null
sequence, so from the shift rule (qn 36(c)) (
a
) is a null
sequence.
