Download has the limit L

Section 8.1: Sequences
Definition:
A sequence is an ordered set of real
numbers that has a one-to-one correspondence
with the positive integers.
a1 , a2 , a3 , a4 , ... , an , ...
• Often denoted by {an}.
• Ex. {2n} = 21, 22, 23, … , 2n, …
• The expression for the nth term is called a
closed-form definition for the sequence.
Definition:
A sequence can also be defined as a
function where the domain is restricted to the
positive integers. In this case f(n) = an.
f(1), f(2), f(3), ... , f(n), ...
The points (1, f(1)), (2, f(2)), (3, f(3)), … can be
discretely graphed on the coordinate plane.
Example 1
List the first 4 terms and the tenth term of each
sequence.
3 4 5
11
a.  n  1 
2, , ,
and a 
 n! 
2


n
n 1
b. (1)

3n  1 

2
6
24
10
10!
1 4 9 16
100
,- , ,and a10  
2 5 8 11
29
A recursively defined sequence will state the
first term, a1, and then define ak+1 in terms of the
preceding term ak.
Ex.
Find the first four terms of the sequence
defined recursively by a1 = 2 and ak+1 = 3ak
for k ≥ 1. Can you find a closed form
definition for this series?
2, 6, 18, 54 .
Yes, an = 2(3)n-1 (geometric)
Definition:
A sequence {an} has the limit L, or
converges to L, if for every ε > 0 there exists a
positive number N such that |an – L | < ε
whenever n > N. Denoted: lim an  L
n 
If such a number L does not exist, the sequence
has no limit, or diverges.
Example 2
Let {an} =
a. Find
 1 
 n1 
  2  
lim an
n 
lim
n 
1
 2
n 1
0
b. Let ε = .001. Find some integer N such that
for all n > N an is less than ε units from the
limit above.
N  10
Properties of Sequences
Let {an} be a sequence and f(n) = an. If f(x)
exits for every real number x ≥ 1, then
1) If lim f ( x)  L , then lim f ( n)  L
x 
n 
2) If lim f ( x)   , then lim f ( n)  
Also,
x 
For any real number r,
n
r  0 if r  1 .
1) lim
n 
2) lim r n   if r  1
n 
n 
Example 3
Determine whether the sequence converges or
diverges. If it converges, find the limit.
a. (1) n1
diverges
b.
c.
 5n 
 2n 
e 

2  
converges to 0

2 n
3
converges to 2
More Properties of Sequences
Sandwich 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 
and
Theorem 8.8:
Let {an} be a sequence.
If lim an  0 , then lim an  0
n 
n 
Example 4
Determine whether the sequence converges or
diverges. If it converges, find the limit.
a. (1)n 1 1 
converges to 0

b.
 cos 2 n 
 n 
 3 
n
converges to 0