Math 104 – Rimmer
12.1 Sequences
12.1 Sequences
A __________________ is an ordered list of numbers.
A sequence can be _____ or _________.
In this class we will deal
with _______ sequences
Note: the sequence
doesn’t have to
start at n = 1
Notation:
{a1 , a2 , a3 ,… , an , an+1 ,…}
{a n }
or
∞
{an }n =1
a formula for the nth term
3
4
5
1 2

,
,
,
,…
 ,
 4 9 16 25 36 
an =
Math 104 – Rimmer
12.1 Sequences
n
( n + 1)
2
input :
output :
 1  2  3   4   5 
1,  ,  2,  ,  3,  ,  4,  ,  5,  ,…
 4   9   16   25   36 
These isolated points make up the graph of the sequence.
It seems as though the terms of the
sequence are approaching ____ as n → ∞
lim
n →∞
lim
n →∞
n
( n + 1)
p (n)
q (n)
2
=0
( p and q polynomials )
when _____________________ .
1
Math 104 – Rimmer
12.1 Sequences
In general, if the terms of the
sequence are approaching L as n → ∞, then
lim an = L
n →∞
When this limit exists and is finite, we say the sequence is __________ .
When the lim an does not exist or is infinite, the sequence is called ________ .
n →∞
  nπ
cos 
  2



  nπ
⇒ cos 
  2

 nπ 
cos 
  is ______ since the lim
 __________.
n →∞

 2 
 n2 


n + 2
n2
=∞
n →∞ n + 2
lim
 n2 
⇒
 is __________.
n + 2
lim
n →∞
p ( n)
q ( n)
=∞
( p and q polynomials )
when ___________________.
Math 104 – Rimmer
12.1 Sequences
So basically, finding the limit of a sequence
boils down to being able to find limits at infinity.
Tools :
Section 2.2 Limit Laws
Section 4.4 Limits at Infinity
Section 7.8 Indeterminate forms and L'Hopitals Rule
Thoerems :
1. Squeeze Theorem:
an ≤ cn ≤ bn for all n > N 

and
cn = L
 ⇒ lim
n →∞
lim an = lim bn = L 
n →∞
n →∞

2. lim an = 0 ⇒ lim an = 0
n →∞
n →∞
4. lim an = L 
n →∞

and
f ( an ) = f lim an = f ( L )
 ⇒ lim
n →∞
n →∞
bring the limit inside
f is contin. at L 

(
)
5. Every bounded and increasing
sequence and every bounded and
decreasing sequence is convergent.
if − 1 < r ≤ 1
convergent
3. The sequence r n is 
 divergent for all other values of r
0 if − 1 < r < 1
lim r n = 
n →∞
if r = 1
1
{ }
2
Math 104 – Rimmer
12.1 Sequences
Determine whether the sequence converges or diverges. If it converges, find the limit.
3 + 5n 2
an =
n + n2
1
an =  
π 
n
Determine whether the sequence converges or diverges.
If it converges, find the limit.
an =
an
Math 104 – Rimmer
12.1 Sequences
n +1
9n + 1
( −1)
=
n −1
n
n2 + 1
3
Determine whether the sequence converges or diverges.
Math 104 – Rimmer
12.1 Sequences
If it converges, find the limit.
 n +1 
an = 

 n 
n
Math 104 – Rimmer
12.1 Sequences
4
Math 104 – Rimmer
12.1 Sequences
Determine whether the sequence converges or diverges.
Math 104 – Rimmer
12.1 Sequences
If it converges, find the limit.
an =
( −1)
n
( )
sin n 2
n
5