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SECT. 9-1 SEQUENCES
Sequence
An ordered collection of numbers in a
prescribed order defined by a function
f(n)
The values, an are called terms
a1, a2, a3, a4 ,..., an ,...
notation { a1, a2, a3 , a4 , ...}  {an }  {an }n 1


1) Write the first five terms of the sequence
n

n 1
a)
an 
b)
1 (n  1)

an 

n
3


n
Defining Sequences
Determine the pattern in the sequence
And use pattern to determine the nth
term using inductive reasoning
2 7 14 23
1, , ,
,
...
4 9 16 25
12  2 2 2  2 32  2 4 2  2 5 2  2
,
,
,
,
...
2
2
2
2
2
1
2
3
4
5


n2  2
an 
n2
2) Write the next two terms for the given
sequence
7
9
, 4, , 5 ....
2
2
rewrite : 3.5, 4, 4.5, 5, 5.5, 6, ...


A recursively defined sequence: given the
first term, all other terms are defined using
that term. d1 = 3.5 and {dn}= dn+1
3) Find next three terms:
pattern?
5, 10, 20, 40, ...

4) Find the general term an
3

4
5
6
7
, ,
...
 , - ,
5
25 125
625 3125 
Numerator: start with 1 for first term, add 2

Denominator: powers of 5
Notice terms alternate signs
(1) n1 or (1) n
5) Write and expression for the nth term

4 5 6 
2, 1, , , ,...

5 7 9 
rewrite
2 3 4 5 6 
 , , , , ,...
1 3 5 7 9 
n 1
an 
there may be several ways to write the nth term
 2n 1
Convergence and Divergence
of a sequence
We say that a sequence converges
to a limit L if
lim
an   L
n 
If no limit exists then an diverges.
If the terms increase without bound,

{an} diverges to infinity
Convergence or Divergence?
Convergence and Divergence ?
lim
n 

an   L
Properties of Sequences
if lim an  L and
n
lim bn  K then
n
1. lim an  bn   L  K
n

2. lim can  cL

n
3. lim an bn   Lk


n
an L
4. lim
 , bn  0 and K  0
n b
K
n
6) Evaluate
lim
1
n  n

n
lim
1 1

 
n n 
Squeeze Theorem for Sequences:
If {an}, {bn}, and {cn} are sequences and
an ≤ bn ≤ cn for every n and
if lim an  L  lim cn , then lim bn  L
n 
n 
n 
7) Determine whether the sequence converges or
diverges. If it converges, find the limit.
a.
b.
c.
n1
(

1)


 5n 
 2n 
e 
2     
2 n
3
8) Determine whether the sequence converges
or diverges. If it converges, find the limit.

n 1 1 
a. (1)

n

b.
c.
 cos 2 n 
 n 
 3 
1
n
lim
2
n! 

1 n n   n!
2
9) Determine whether the sequence converges
or diverges. If it converges, find the limit
3  5n 2
an 
n  n2
lim 3  5n 2

2
n  nn
Factorial (!)
5! = 5 4  3 21
10) simplify

11) simplify

25!
23!
n  2!
n!
Increasing and Decreasing Sequences
•A sequence {an} is increasing if
an  an 1 for all n  1
a1  a2  a3  ...
•A sequence {an} is decreasing if


an  an 1 for all n  1
a1  a2  a3  ...
•A sequence {an} is monotonic if it is either
always increasing or always decreasing
Monotonic? Sequences
Bounded Sequences
•A sequence {an} is bounded from above if
there is a number M such that an  M for all n
•A sequence {an} is bounded from below if


there is a number M such that an  M for all n
Bounded Sequences
•A sequence {an} is called Bounded
if it is bounded either from above or below.
•If a sequence {an} is bounded
and monotonic,
then it converges.
12) Is the sequence {an} bounded?
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