Notes on Sequences
Definition
: A sequence
is an ordered list of numbers:
, , , ,
a b c d
K
. Each number is called a
term of the sequence
and is referred to by the number of its position in the listing. So in
the listing above,
a
is the first term,
b
is the second term,
c
is the third term and so forth.
If there are only a finite number of terms in the sequence, then the sequence is called a
finite sequence
. Otherwise there are an infinite number of terms in the sequence and the
sequence is called an infinite sequence
. This course deals almost exclusively with infinite
sequences. So unless otherwise stated, the term ‘sequence’ means an ‘infinite sequence’.
Notation
: When discussing sequences we need to have notation that is compact yet
complete. Therefore we employ the following notation to indicate a sequence:
{
}
1
2
3
4
,
,
,
,
,
,
n
a a a a
a
K
K
or in a more compact form
{
}
1
n
n
a
∞
=
.
In this second form, the role of
n
is that of a counter
and indicates the order in
which the terms are listed. It is called the index
for the sequence. Notice that in this case
we have “counted” the terms of the sequence corresponding to the positions they hold.
That is to say,
1
a
is the first term,
2
a
is the second term,
3
a
is the third term, and so forth.
In general,
n
a
is the
th
n
term. However it is sometimes desirable to change the index. We
could, for example, write the above sequence as:
{
}
1
0
n
n
a
∞
+
=
or
{
}
1
2
n
n
a
∞

=
.
In both cases the terms of the sequence are the same and listed in the same order. So they
are the same sequence. In fact, if we have two sequences, say
{
}
1
n
n
a
∞
=
and
{
}
i
i k
b
∞
=
, that are
the same numbers listed in the same order, we say the sequences are equal
and write
{
}
{
}
1
n
i
n
i k
a
b
∞
∞
=
=
=
.
Finally, there will be occasions in which the counter is unimportant to the
discussion. In these cases we simply write
{
}
n
a
.
Examples