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11.1
Sequences
Sequence
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.
Examples
n
an 
n2
an 
 1
n
n 1
an  5
2
{1, 4,9,16, 25,...}
{1, 1, 2, 3, 5, 8, 13, 21,...}
The last example is a recursively defined
sequence known as the Fibonacci Sequence.
Limit and Convergence
n
• Let’s take a look at the sequence an 
n2
• What will happen as n gets large?
• If a sequence {an} approach a number L as n
approaches infinity, we will write
lim an  L
n 
and say that the sequence converges to L.
• If the limit of a sequence does not exist, then the
sequence diverges.
Example
2n  1
Does an 
converge?
n
2n  1
 2n 1 
 lim 
 
lim
n

n
n
 n n
2n
1
 lim
 lim
n  n
n  n
 20
2
The sequence converges to 2.
Graph the sequence.
Properties of Limits
• Same as limit laws for functions in chapter 2.
• Theorem:
Let f (x) be a function of a real variable such that
lim f ( x)  L
x 
If {an} is a sequence such that f (n) = an for every positive
integer n, then
lim a  L
n 
n
• Squeeze Theorem
• Absolute Value Theorem:
For the sequence {an},
if nlim
a  0, then nlim
a  0.
 n
 n
Examples
Determine the convergence of the following sequences.
3
an 
n
sin( 2n)
bn 
1 n
n 1
an 
3n  2
bn  (1) n!
ln n
an  3
n
n
 3
cn  

 5 
n
Monotonic Sequence
• A sequence is called increasing if an  an 1
for all n.
• A sequence is called decreasing if an  an 1
for all n.
• It is called monotonic if it is either increasing
or decreasing.
Bounded Sequence
• A sequence is bounded above if there is a
number M such that an ≤ M for all n.
• A sequence is bounded below if there is a
number N such that N ≤ an for all n.
• A sequence is a bounded sequence if it is
bounded above and below.
Theorem: Every bounded monotonic sequence is convergent.
Examples
Determine whether the sequence is bounded, monotonic
and convergent.
an  n!
6
an  2
n
n
bn 
n 1
an  (2)
n
an  ( 1)  1
n
3
n
cn  4
n  1000
A geometric sequence is a sequence in which the ratios
between two consecutive terms are the same. That same
ratio is called the common ratio.
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
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