Sequences
An Infinite Sequence is an unending succession of terms with a DEFINITE order. That is,
1st term, 2nd term, 3rd term, 4th term, ……., nth term, (n+1)th term, (n+2)th term, …..
*** We want to be able to use the pattern to find the nth term of the sequence. ****
Example: In each of the following, find the nth general term of the following sequence:
(A) 3, 6, 9, 12, ….
(B)
(C)
(D)
(E) 1, 5, 1, 5, 1, 5, ….
Sequences Written Page 1
Sequence Graphically
A sequence is a function whose domain is a set of integer values, so
is alternative notation for the function
for values of x = 1, 2, 3, 4, ….
Based on this information, we can talk about the graph of a sequence which consists of ISOLATED POINTS.
For example, consider the sequence given by
means we can graph
** we see that as
,
Graphically, we say the sequence _______________,
and we write,
For example, consider the sequence given by
** we see that as
,
Graphically, we say the sequence _______________,
and we write,
Sequences Written Page 2
Sequence Graphically
For example, consider the sequence given by
** we see that as
,
Graphically, we say the sequence _______________,
and we write,
For example, consider the sequence given by
** we see that as
,
Graphically, we say the sequence _______________,
and we write,
Sequences Written Page 3
Sequences Graphically
For example, consider the sequence given by
** we see that as
,
Graphically, we say the sequence _______________,
and we write,
For example, consider the sequence given by
** we see that as
,
Graphically, we say the sequence _______________,
and we write,
Sequences Written Page 4
Limit of a Sequence Algebraically
A sequence
approaches a
,
eventually become arbitrarily close to the
Definition: A sequence
if the
, if the
of the sequence
.
is said to CONVERGE to the limit, L,
. Basically, as n tends to infinity, if the limit approaches a finite
number, then we say the sequence CONVERGES.
Otherwise, the sequence diverges (does not near a finite value).
Example: Do the following sequences converge or diverge?
(A)
(C)
(B)
Sequences Written Page 5
Examples to determine whether a sequence converges or diverges
Do the following sequences converge or diverge?
(A)
(B)
(C)
Sequences Written Page 6