Graphics Reference
In-Depth Information
a
n
n
2 It is possible to draw a graph of the sequence (
a
) by plotting the
points (
n
,
a
) on a conventional cartesian graph of R
R.
Draw the graphs of the first five terms of the six sequences given
above.
Although graph drawing provides a useful picture of the first few terms
of the sequence, it does not illustrate what happens in the long run.
Most of the important questions about sequences, which go on for ever,
are not decided by the first few terms but depend on the behaviour of
the
n
th term
a
as
n
gets larger and larger, and these terms are not
illustrated on the graphs you have drawn.
3
(a) According to the definition at the start of this chapter, why
do the integers from 1 to 10, in order,
not
form a sequence?
Or the integers from 1 to a million, in order?
(b) Construct a sequence whose terms take only one value. Such
a sequence is called a
constant sequence
.
(c) Construct a sequence whose terms take exactly three values.
(d) Construct a sequence whose terms,
a
, take exactly three
values but which becomes constant for large
n
.
The central property of sequences we will study is that of convergence.
This turns out to be surprisingly diMcult to define. In order to become
familiar with sequences, we first examine a number of more
straightforward properties which we will find useful.
Monotonic sequences
4 Test each of the sequences (i)
—
(vii), defined below, to determine
whether any one or more of the following four properties, (a)
—
(d),
holds for all values of
n
.
