Graphics Reference
In-Depth Information
53 Let (
a
) be a sequence.
(i) Weknow that
a
0 for
n
10
, what do you think thelimit
is?
(ii) Weknow that
a
1 for
10
n
10
, what do you think thelimit is?
(iii) The sequence is given by
a
0 for
n
10
, and
a
0 for
n
10
,
a
1 for
10
n
10
, and
a
2 for 10
n
. What is thelimit?
54 Let (
a
)
a
and (
b
)
b
.
(i)
The scalar rule
Use the scalar rule for null sequences (qn 32) to show that
(
c
·
a
c
·
a
.
(ii)
The subsequence rule
Use the subsequence rule for null sequences (qn 36(a)) to show
that every subsequence of (
a
)
) converges to
a
.
(iii)
The sum rule
Use the sum rule for null sequences (qn 44) to show that
(
a
)
a
b
.
(iv) Usethescalar ruleand sum ruleto show that
(
c
·
a
b
d
·
b
)
c
·
a
d
·
b
.
(v)
The difference rule
Apply (iv) to show that (
a
b
)
a
b
.
(vi)
The product rule
Usethescalar rul, thesum ruleand theproduct rulefor null
sequences (qn 43) and the equation
a
b
ab
(
a
a
)(
b
b
)
a
(
b
b
)
b
(
a
a
)
)
ab
.
(vii)
The absolute value rule
Use the absolute value rule for null sequences (qn 33), the
reverse triangle inequality (qn 2.63) and the squeeze rulefor
null sequences (qn 34) to show that (
to show that (
a
b
a
)
a
.
(viii)
The squeeze rule or sandwich theorem
When
a
, for
n
k
, and
a
b
, use the squeeze rule
for null sequences (qn 46(ii)) to show that (
c
c
b
)
a
.
55
(i) Extend the sum rule (qn 54(iii)) to a sum of three convergent
sequences and then to a sum of
k
convergent sequences.
(ii) Extend the product rule (qn 54(vi)) to a product of three
convergent sequences, and then to a product of
k
convergent
sequences.
(iii) If (
a
)
a
, what is thelimit of thesequence
(1
2
a
3
a
4
a
5
a
)?
