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From qn 3.51 write down a sequence of terminating decimals which
converges to
.
The limit of the sequence 0.3, 0.33, 0.333, 0.3333, . . .(with a
single
recurring digit) is denoted by 0.3
, or 0.33
, or 0.333
.
Wehaveproved that 0.3
.
12 Provethat the
n
th term in the sequence 0.9, 0.99, 0.999, 0.9999, . ..
is equal to 1
1/10
. Deduce that 0.9
1.
13 For the sequence 0.1, 0.13, 0.131, 0.1313, . . . the 2
n
th term is
).
Use 1.3(vi) to simplify this expression, and prove (with the help of
3.80) that the sequence converges to
(1
10
10
...
10
.
This proves that 0.1
3
or 0.131
3
(either of which denotes the limit of
the sequence)
. Here there is
a pair of
recurring digits.
14 Useqn 1.3(vi) and 3.39 to provethat thelimit of thesequencewith
n
th term
1
10
1
10
...
1
10
a
b
is
a
b
/999.
Find terminating decimals
a
and
b
such that 12.456
7
8
a
b
/999.
Here there is
a triple of
recurring digits.
Must every infinite decimal with a recurring block of digits be
equal to a rational number?
The
infinite decimal
d
.
d
d
d
...
d
. . . is thelimit of theinfinitedcimal
sequence with
n
th term
d
d
d
10
10
d
10
...
,
where
d
is an integer, and, when 1
i
,
d
is either 0, 1, 2, . . . or 9. (This
definition looks a bit strange when
d
is negative, but it is convenient
for proving theorems to have all infinite decimal sequences increasing.)
This definition does not make the claim that every infinite decimal
sequence is necessarily convergent or that every infinite decimal is
necessarily a number.
In question 3.51 we found that every number on the number line is the
limit of an infinite decimal sequence.
When for all suMciently large
n
,
d
0, thedcimal is said to
terminate
.
When there is a number
b
such that for all suMciently large
n
,
d
d
, thedcimal is said to
recurr
(with a recurring block of length
b
).
15 Show that every rational number which is not equal to a