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lower bound for (
a
), the terms of the sequence are non-negative.
Use3.75.
(viii)
t
is a lower bound for (
a
) while
t
1/10
is not, so there are
terms of the sequence (
a
) between these numbers,
e
.
g
.
t
D
a
t
1/10
D
1/10
, for somevalueof
n
and
) is decreasing, for all subsequent
n
.
(ix) (viii) implies that 0
since(
a
a
D
1/10
for suMciently large
n
. Given
0,
1/10
for some
i
,so(
a
)
D
by definition.
35 If (
a
) is monotonic increasing and bounded above, then (
a
)is
monotonic decreasing and bounded below. From qn 34 (
a
)
A
, say,
A
.
so (
a
)
36 The sequence is convergent from qn 35. The limit is between 2 and 3
by the closed interval property, qn 3.78. In qn 11.32 we will see that
thelimit is .
37
a
x
a
/
x
x
a
/
x
(
x
a
/
x
)
x
(
(
x
a
/
x
))
x
and
0
(
x
a
/
x
)
a
(
(
x
a
/
x
))
.(
x
) is monotonic decreasing and
bounded. Let (
x
l
. Apply thesum rule(3.54), thescalar rule(3.54)
and therciprocal rule(3.65) to obtain
l
)
(
l
a
/
l
), and thus
l
a
.
Because the terms are positive
l
a
.
38 (
a
) is decreasing, so both are convergent by qns 34
and 35. For the sequences of 2.38, 2.38(iii) implies that (
b
) is increasing, (
b
)isa
null sequence, so by the difference rule the sequences have the same
limit. In the case of the sequences of 2.39, the property
b
a
a
(
b
a
) implies that
b
a
(
)
(
b
a
) and
so (
b
a
) is a null sequence, and the sequences have the same limit.
39 If
a
exists for all positive real
a
, then
a
etc. exist.
40
(i) (
a
) is increasing and bounded above.
(ii) (
b
) is decreasing and bounded below.
a
a
(iii)
b
(
)
(
b
), so the difference rule makes the limits
equal.
(iv) Repeated use of product rule.
(v) Difference rule.
(vi) 0
c
a
b
a
.
(vii) By difference rule.
41
(i) The sequence is decreasing and bounded below by 0, so it is
convergent by qn 34.
(ii)
n
(
a
)/
a
.
n
(1
(1/
a
)
1)
(
a
1)ยท
b
(
b
1).
In chapter 11 we will find that lim
n
(
a
1)
log
(iii)
(
ab
)
1
a
.
42
(i)
a
a
...
a
b
...
b
b
.
