Graphics Reference
In-Depth Information
Boundedness of convergent sequences
61 For what values of
L
and
U
is theinequality
L
x
U
equivalent
to theinequality
x
a
? Mark
a
,
,
L
and
U
on a number
line.
For these values of
L
and
U
theintrval
x
L
x
U
is called
an
-
neighbourhood
of
a
.
62 Let (
a
) is eventually bounded
aboveby
a
1 and eventually bounded below by
a
1. (
Hint
.
Take
1.) Illustratethis proof with a graph.
Deduce from qn 13 that every convergent sequence is bounded.
Give an example to show that a bounded sequence need not be
convergent.
)
a
. Prove that the sequence (
a
63 Let (
a
0. Identify a positive number which is
eventually a lower bound for the sequence (
a
)
a
, and
a
).
Quotients of convergent sequences
64 What is the relationship between the limits of the sequences
(2
n
/(
n
1)) and ((
n
1)/2
n
)?
65
The reciprocal rule for non
-
null sequences
Let (
a
) be a sequence of non-zero terms with (
a
)
a
and
a
0.
Wewish to show that (1/
a
)
1/
a
, and so wemust examine
1
a
1
a
a
a
1
a
ยท
a
a
a
.
a
a
a
a
(a) How do you know that
is a null sequence?
a
2
a
1
x
y =
1
a
2
3
a
a
2
3
a
2
a
(b) As in qn 63, show that, eventually,
a
2
a
3
a
2
,
