Transcript Calculus 8.1 - University of Houston

8.1: Sequences
Craters of the Moon National Park, Idaho
Photo by Vickie Kelly, 2008
Greg Kelly, Hanford High School, Richland, Washington
A sequence is a list of numbers written in an explicit order.
an   a1, a2 , a3, ... , an , ... 
nth term
Any real-valued function with domain a subset of the
positive integers is a sequence.
If the domain is finite, then the sequence is a finite sequence.
In calculus, we will mostly be concerned with infinite sequences.

A sequence is defined explicitly if there is a formula that
allows you to find individual terms independently.
an
Example:
1


n
n2  1
To find the 100th term, plug 100 in for n:
 1
100
a100
1


1002  1 10001

A sequence is defined recursively if there is a formula that
relates an to previous terms.
Example:
b1  4
bn  bn1  2 for all n  2
We find each term by looking at the term or terms before it:
b1  4
b2  b1  2  6
b3  b2  2  8
b4  b3  2  10
You have to keep going this
way until you get the term you
need.

An arithmetic sequence has a common difference
between terms.
Example: 5, 2, 1, 4, 7, ...
ln 2, ln 6, ln18, ln 54, ...
Arithmetic sequences can
be defined recursively:
or explicitly:
d 3
6
d  ln 6  ln 2  ln
 ln 3
2
an  an1  d
an  a1  d  n 1

An geometric sequence has a common ratio between
terms.
Example: 1, 2, 4,  8, 16, ...
102 , 101 , 1, 10, ...
Geometric sequences can
be defined recursively:
or explicitly:
r  2
101
r  2
10
 10
an  an1  d
an  a1  d n1

If the second term of a geometric sequence
is 6 and the fifth term is -48, find an explicit
rule for the nth term.
Example:
a1  r
48

a1  r
6
4
r 3  8
r  2
a2  a1  r 21
6  a1   2
3  a1
an  3  2 
n 1

Sequence Graphing on the Ti-89
Change the graphing mode to “sequence”:
MODE
Graph…….
4
ENTER

n 1
an   1
n
n
Example: Plot
Y=
Use the alpha key to
enter the letter n.
Leave ui1 blank for
explicitly defined
functions.

WINDOW

WINDOW
GRAPH

The previous example was explicitly defined.
Now we will use a recursive definition to plot
the Fibonacci sequence.
a1  1
Y=
a2  1
an  an2  an1
Use the alpha key to
enter the letters u
and n.
Enter the initial values
separated by a
comma (even though
the comma doesn’t
show on the screen!)

Enter the initial values
separated by a
comma (even though
the comma doesn’t
show on the screen!)
WINDOW

WINDOW
GRAPH
You can use F3 Trace
to investigate values.

We can also look at the
results in a table.
TBLSET
TABLE
Scroll down to see
more values.

TABLE
Scroll down to see
more values.

You can determine if a sequence converges by finding the
limit as n approaches infinity.
2n  1
Does an 
converge?
n
2n  1
lim
n
n
 2n 1 
lim   
n
 n n
2n
1
lim
 lim
n n
n n
20
2
The sequence converges and its
limit is 2.

Absolute Value Theorem for Sequences
If the absolute values of the terms of a sequence converge
to zero, then the sequence converges to zero.
Don’t forget to change back to function mode
when you are done plotting sequences.
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