Graphics Reference
In-Depth Information
36
(a) Must every subsequence of a null sequence be null? (The
answer here gives
the subsequence rule for null sequences
)
(b) Check the converse. If a sequence has a subsequence which is
null, must the original sequence have been null?
(c)
T
he shift rule for null sequences
For some fixed positive integer
k
, the sequence (
a
) is null.
Provethat (
a
) is null. The sequence (
a
) is a special kind of
) for which theconvrseof thersult in (a)
is
valid. The result may also be expressed by saying that a
sequence which is eventually null is a null sequence.
(d)
A squeeze rule for null sequences
,
with a shift
I
Let (
a
subsequence of (
a
b
a
, for all
n
k
.
) be a null sequence, and 0
Provethat (
b
) is a null sequence.
37 Prove that each of the following sequences is null by building on
qn 36:
(i) (1/(2
n
1)),
(ii) (1/(2
n
1)),
(iii) (
a
), where
a
n
for
n
10, and
a
1/
n
for
n
10,
(iv) (1/(3
n
100)).
38
(i) If a constant sequence (
c
) is a null sequence, what can be said
about
c
?
(ii) If two numbers
a
and
b
havetheproprty that
a
b
1/
n
for all positive
n
, what can besaid about thenumbrs
a
and
b
?
39 A particularly important family of null sequences is formed by
geometric progressions. Suppose 0
x
1.
(a) Provethat 1
1/
x
, using chapter 2 qn 23, and let
1/
x
1
y
,so
y
0.
(b) Show that (1/
x
)
1
ny
, using Bernoulli's inequality,
chapter 2 qn 29.
(c) Provethat 0
x
1/(
yn
1).
(d) Use1/(
yn
1)
(1/
y
)(1/
n
), thescalar ruleand thesquzerule
to provethat (
x
) is a null sequence.
(e) Prove that for any constant
c
,(
c
ยท
x
) is a null sequence.
40
(a) Use a calculator or a computer to evaluate terms of the
sequence (
n
/2
).
Calculatethevalus for
n
1, 2, 5, 10, 20, 50.
Would you conjecture that the sequence is null?
. Verify that
a
(b)
Let
a
n
a
/2
(1
)
.
