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In-Depth Information
Sequences tending to infinity
17 Describe the following four sequences saying whether or not they
are monotonic and whether or not they are bounded above or
below.
(i) 1, 1, 2, 1, 3, 1, . . .,
n
,1,...
(ii)
1, 2,
3, 4,
5, ..., (
1)
n
,...
(iii) 2, 1, 4, 3, . . ., 2
n
,2
n
1, . . .
(iv) 11, 12, 11, 12, . . ., 11, 12, . . .
In each case decide whether there is a stage beyond which all terms
in the sequence are
(a) greater than 1, in which case we say that the sequence is
eventually greater than 1,
(b) greater than 10, so that the sequence is eventually greater
than 10,
(c) greater than 100, so that the sequence is eventually greater
than 100.
Only sequence (iii) is said to tend to
(plus infinity).
We say that a sequence
tends to
when any number,
C
, however
chosen, is eventually a lower bound for the sequence. So a sequence
must pass infinitely many tests if it is to be said to tend to
because
there are infinitely many choices for
C
, and the terms of the sequence
must eventually exceed each of them.
For each value of
C
, welook for a value
N
of
n
suMciently great to
make
a
C
, for all
n
N
.
18 Select values of
C
to demonstrate that qn 17(i), (ii) and (iv) do not
tend to
.
a
n
C
n
N
