Graphics Reference
In-Depth Information
3
Sequences
A first biteat infinity
Preliminary reading:
Hemmings and Tahta, O'Brien.
Concurrent reading:
Hart.
Further reading:
Knopp.
The notion of a sequence is a familiar and intuitive one. The following
examples of sequences of numbers illustrate this notion.
1, 2, 3, 4, 5, . . .
2, 4, 6, 8, 10, . . .
, ...
3, 3.1, 3.14, 3.141, 3.1415, . . .
1, 1,
1, 1,
1, . . .
,
,
,
,
,
,
,
,
, ...
The general way of writing down a sequence is
a
,
a
,
a
,...,
a
,...
so that for each natural number
n
, there is an element of the sequence,
a
. This means that a sequence must be an infinite (not finite) list of
terms, though repetition is allowed as in the fifth example above. Such
a sequence is denoted by (
a
).
a
is the single number which appears as
the
n
th term of the sequence. (
a
) refers to an infinite list of numbers, in
order; (
a
) is the sequence as a whole.
1 For each of the sequences listed above, except the fourth, suggest a
formula for the
n
th term,
a
. (A sequence may or may not have a
formula to define it.)
