11.1: 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 (>0) 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

A 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  r
an  a1  r n 1

You can find the limit of a sequence by finding the limit as n
approaches infinity. If an = f(n) and if lim f(n) exists as n→∞,
then lim an also exists.
Does an 
2n  1 have a limit?
n
2n  1
lim
n
n
 2n 1 
lim   
n
 n n
20
2
The limit is 2.
2n
1
lim  lim
n n
n n

Convergence/Divergence
an  L means that {a } has a limit, L.
• lim
n
n 
• Sequences that have limits converge.
• Sequences that do not have limits diverge.
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.
p
Sequence Patterns, Monotonicity,
Boundedness, and Convergence
Write explicit rules for an for the following
sequences.
• {3, 7, 11, 15}
• {1, -1/4, 1/9, -1/16}
• {3/2, 4/5, 5/8, 6/11, ½}
•{1, x, x2/2, x3/6, x4/24, x5/120}
Monotonicity
• Def. {an} is monotonic if its terms are
nondecreasing or nonincreasing.
a1  a2  a3  ...  an  ...
a1  a2  a3  ...  an  ...
Boundedness
• {an} is bounded above if  M| an  M  n.
• {an} is bounded below if  N| an  N  n.
• {an} is bounded if it is bounded above and
below.
• Thm. {an} converges if it is monotonic and
bounded.
4 ways to determine
monotonicity
•
•
•
•
Test actual numbers
Analyze the first derivative of an
Compare an and an+1
Graph the sequence
Determine monotonicity
Ex.
3n
an 
n2
3
an   
2
n
(See next slides for solution.)