Graphics Reference
In-Depth Information
49 Prove
(i) (
n
/(2
n
1))
;
(ii) ((2
n
1)/(
n
1))
2.
50 The sequence ((
1
and
1. The definition of a limit rules out ambiguity. This is quite
easy to prove. Suppose (
a
1)
), for example, does not have two limits,
) is a sequence and both (
a
)
a
and
(
a
b
)isa
null sequence. Now use the difference rule (qn 45) to show that the
constant sequence (
b
a
) is a null sequence and deduce that
a
b
(qn 38). In fact, the sequence ((
1)
) is not convergent.
)
b
. By definition (
a
a
) is a null sequence and (
a
51 Look back to qn 19 and the discussion preceding it, for the
definition of [
x
]. Write down the first four terms of the sequence
([10
a
]/10
)
(i) when
a
6
,
(ii) when
a
,
(iii) when
a
2.
Provethat, for
any a
,
[10
a
]
10
1
10
0
a
.
Deduce that for any number
a
there is a sequence of rational
numbers tending to it.
This question shows both the power and the limitation of decimal
expressions. Each of the terms of the sequence ([10
a
]/10
)isa
terminating decimal
and if the sequence is eventually constant, then the
number
a
is a
terminating decimal
. Terminating decimals provide an
excellent way of approximating to the points of a number line, but
there are many points on a number line which may only be pinpointed
as the
limit
of a sequence of terminating decimals and not by a single
terminating decimal. An infinite decimal is the limit of a sequence of
terminating decimals, presuming that sequence converges. The infinite
decimal 1.4142. . . denotes the limit of the sequence 1, 1.4, 1.41, 1.414, . . .
and this is what is meant by writing
2
1.4142 . . .. The sequence
determines a unique number
—
the limit of the sequence.
Many of the theorems we have established for null sequences can
be extended to give theorems about convergent sequences whatever
their limits.
52
The shift rule
Use the shift rule for null sequences (qn 36(c)) to prove that if
(
a
)
a
. This result
may also be described by saying that a sequence which eventually
converges to
a
, converges to
a
.
)
a
for some fixed positive integer
k
, then (
a
