Graphics Reference
In-Depth Information
70 (
n
/2
) for any positive integer
k
. All null.
Take
, then for suMciently large
n
,
a
/
a
.
71 Take
(1
l
), then 0
a
(
(1
l
))
a
.
0
l
1
0
(1
l
)
1, so qn 39 can be applied here.
72 Take
(
l
1), then (
(1
l
))
a
a
.1
l
1
(1
l
).
73 (1/
n
). (
n
).
74 The sequences are null if and only if the sequences of absolute
values are null.
(i) null when
x
1, or when
x
1 and
k
0.
(ii) null for all values of
x
.
75 (
a
a
) is a null sequence, so by the squeeze rule (qn 34), (
a
)is
null, so
a
0. Thus for a sequence of non-negative terms, a
negative limit is impossible.
76 By the difference rule (qn 53(v)), (
b
a
) has limit
b
a
. But
(
b
a
) is a sequence of non-negative terms, so
b
a
0.
78 Since0
a
A
,0
a
A
, from qn 75.
Since0
B
a
,0
B
a
, from qn 75. 0
1/
n
2.
79
a
a
(
a
2)
(
a
2)
(
a
a
)(
a
a
)
a
a
a
. Since the terms are positive,
a
has thesame
a
sign as
a
.
80
(a) If
A
a
A
, then the monotonic increasing
subsequence gives
n
n
A
a
a
A
, while
convergence to
A
gives
a
A
for suMciently large
n
.
(c) Convergence to
A
implies that
a
A
for suMciently
large
n
,or
A
a
A
, so from (a)
A
a
A
.
(d) Take
N
as a suMciently large
n
in (c). Then claim monotonic
increasing and (b).
Theorem
. If a monotonic increasing sequence has a subsequence
converging to
A
, then the sequence tends to
A
.
If (
a
) is monotonic decreasing with a subsequence convergent to
A
,
then (
a
) is monotonic increasing with a subsequence converging
to
A
. By the theorem we have proved, (
a
)
A
, and so by
thescalar rul, (
a
)
A
. The overall theorem is that if a monotonic
sequence has a convergent subsequence, then the whole sequence is
convergent to the same limit.
81
(a)
p
p
.
(b)
A
A
.
(c) (
p
) is constant, (
A
) is null.
