Graphics Reference
In-Depth Information
It was mentioned in chapter 2 that many square roots, cube roots, etc.,
of integers are not rational numbers. So if we only knew about rational
numbers, the square root of 2 'would not exist'. Roots of numbers are
problematic. For example,
(
1) 'does not exist' if we are thinking
only of real numbers, as we are in this topic. In chapter 4 we will meet
this issuehead-on and show that
n
th roots of positivenumbrs always
exist and are unique, something which you probably always took for
granted. In the mean time you should proceed as if this is not a
problem.
56 Use a calculator to explore the sequences (
2), (
10) and (
1000).
Repeated use of the square root button gives a subsequence in each
case.
b
.
57
(a) Let 1
b
. Useqn 2.20 to show that 1
b
. Use Bernoulli's inequality (qn 2.29) to show
(b) Let 1
a
that 1
na
b
.
) is a null sequence.
(d) Deduce that (
b
)
1as
n
.
(c) Provethat (
a
(a) Do you think the sequence (
(2
3
)) is convergent?
(b) Use a calculator to explore the sequence (
58
(2
3
)).
(
a
b
)) when 0
a
b
.
(c) Find thelimit of thesequence(
n
(a) For
n
59
2, check that
1. See qn 2.51.
n
1
a
(b) Let
. Use the binomial theorem to show
n
(
n
1)
2
n
(
a
)
.
(c) Show that
2
n
1
, and deduce that (
a
a
) is a null sequence.
(d) Deduce that (
n
)
1.
We say (
a
n
) tends to the limit
a
, or converges to
a
, and write (
a
n
)
a
as
n
if and only if, given any
0, there exists an
N
such that
n
N
a
n
a
.
60 Show that the two definitions of convergence are equivalent.
