11.1 Sequences
Sequence – a list of numbers written in a definite order.
We will only be concerned with
infinite
sequences. An infinite sequence of real numbers is an ordered, unending list.
Some notation for infinite sequences:
{ } {} {} {}
123
11
,,,
nn
n
aaa
a
a
a
∞∞
=
K
n
Often an infinite sequence has a defining function. Some examples are shown below.
1
1
1
1
1
1
1,
,
,
,
,
234
sin
sin
1,0,
1,0,
,sin
,
22
n
n
a
n
a
ππ
∞
∞
=
=−
LL
KK
2
n
π
≥
All sequences do not have a defining function.
It may difficult or impossible to given an explicit
formula for the
n
th term in a sequence
Example:
(1) Sequence of prime integers.
(2) The sequence whose
th term is the
th decimal digit of the number
π.
(3) Fibonacci sequence is defined recursively
12
1
1
1
3
1,1, 2, 3, 5, 8,13, 21, 34, 55, .
..
n
fff
f
f
n
−+
===
+
Sequences can be graphed on your calculator using the parametric mode.
Example: Graph the sequence
1
1
n
n
∞
+
and then look at the terms of the sequence on your
calculator.