Graphics Reference
In-Depth Information
and hence
2
3
a
1
a
2
a
.
(c) Deduce that, eventually,
1
a
1
a
2
a
ยท
a
a
.
a
(d) Now use (a), the scalar rule for null sequences (qn 32) and the
squeeze rule with a shift (qn 46(ii)) to prove that (1/
a
)
1/
a
.
66 Let (
a
) be a sequence of non-zero terms with (
a
)
a
and
a
0.
Provethat (1/
a
)
1/
a
, by applying thersult of qn 65 to the
sequence (
a
).
67
The quotient rule
Let (
a
)
a
,(
b
)
b
, and supposethat all of the
b
and
b
are
a
/
b
.
non-zero. Prove that (
a
/
b
)
68 Find the limits of the sequences with
n
th terms given here, stating
which rules you are using.
(i)
1
1/
n
(ii)
3
n
2
(iii)
n
1
(iv)
(
)
1
2
1/
n
,
4
n
3
,
3
n
n
,
)
1
,
(
(v)
2
1
(vi)
n
n
n
1
3
n
5
(vii)
(
n
3)(
n
1)
3
n
5
n
2
1
,
,
,
(viii)
a
1
a
1
,0
a
, (ix)
1
2
...
n
(x)
b
, where 0
b
1.
,
n
69 Let
a
(
n
8)
n
,
b
(
n
n
)
n
and
c
(
n
n
/8)
n
.
Show that
a
when
n
64.
Find the limits of the sequences (
a
b
c
), (
b
) and (
c
) if they exist.
(
Hint
. For positivenumbrs,
x
and
y
,
x
y
(
x
y
)(
x
y
).
d'Alembert's ratio test
70 Give examples of sequences (
a
) of positivetrms for which
a
a
.
You will find one example in qn 40. Are all the sequences which
you have found convergent? To what limit?