Friday February 4
−
Lecture 14 :
Infinite sequences of numbers
.
(Refers to Section 11.1 in
your text)
Expectations:
1.
Find the limit of simple sequences.
14.1
Definition
A
sequence of numbers
is a function
a
(
i
)=
a
i
whose domain is the set of
positive integers.
It can be denoted as {
a
i
:
i
= 1, 2, 3,.
..} or {
a
1
,
a
2
,
a
3
, .
..}, or simply by
a
1
,
a
2
,
a
3
, .
...
We say that the elements of the sequence are
indexed
with the
natural
numbers.
Its elements are referred to as the
terms of the sequence
, where
a
1
is the first term,
a
2
the
second, and so on.

The order must be respected. Altering the order may alter some of the convergence
properties of the sequence, as we will see later.
14.1.1
Example
The ordered set {
a
i
:
i
= 1, 2, 3,.
..} where
a
i
= 2
i
+ 3 is a welldefined
sequence. Essentially it is the function
f
(
x
) = 2
x
+ 3
with the positive integers as domain.
14.1.2
Definition
A
recursive
sequence is a sequence whose
i
th
term is expressed in terms of
a formula involving previous terms of the sequence.
14.1.2.1
Example
Consider the sequence {
a
i
} defined as
This recursive sequence{
a
i
}
is called a
Fibonacci sequence
.
Some recursive sequences have a "
closed
" form. It is not always easy to determine the
closed form of a recursive sequence, or even determine if it has one. The above Fibonacci
sequence can be shown to have the closed form,
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This closed form describes the
n
th
Fibonacci number given that the first two terms are
a
(0) = 1 and
a
(1) = 1. (We can prove this formula by using linear algebra)
It is useful since we are not required to compute the previous terms to obtain the
n
th
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 Spring '07
 Anoymous
 Calculus, Limit, Natural number, Fibonacci number

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