Section 12-3
Infinite Sequences and Series
Infinite Sequence
• A sequence that has infinite number of terms
is called an infinite sequence.
• We can find the limit of an infinite sequence
by using very large numbers . As the value of n
increases what is happening to the value of a ?
n
Example # 1
• Estimate the limit of the sequence ½, 7/9, 17/28,…
, 2n 2 -1 / n3 +1,…
•
•
lim
n 
2n 2 - 1
n3 + 1
=0
• The denominator grows larger a lot faster than the numerator, thus the
number is becoming smaller and smaller as n gets larger.
Theorems for Limits
Example # 2
• Find each limit.
• 1. lim 3n + 6 = lim (3 + 6/n)= lim
n 
• n  n
n 
2. lim
n 
•
•
•
•
2
n -3n+4
n2
-1
=
lim
n 
3+
lim
n 
6/n = 3+0 = 3
2
n -3n +4
2
n
2
2
n
n - 1
n2
n2
2
n
= 1
=1
1
Limits do not exist for all infinite
sequences
• Find each limit.
2
• lim 4x - 6 no limit because 4x becomes increasingly large 4x/3 - 2/x
• x   3x
•
•
•
•
lim
n
n
(-1) n = (-1) n
n   5n + 1
5n + 1
Find
lim
n
= 1/5
n   5n + 1
n
When n is even (-1) = 1 and when it is odd it will equal -1. Thus the odd numbered
terms approach -1/5 and the even approach 1/5. So there is no limit.
Infinite Series
• An infinite series is the indicated sum of the terms of an infinite sequence.
Sum of an infinite series:
If S is the sum of the first n terms of a series, and S
n
is a number such that S – Sn approaches zero as n
increases without bound, then the sum of the
infinite series is S. lim Sn = S
n 
Sum of an infinite geometric series:
The sum S of an infinite geometric series for which
|r|<1 is given by S= a1
1-r
Important
• In order to have a sum in an infinite series, the
nth term of a series must be approaching 0 as
n approaches infinity. If the nth term does not
approach 0 as n approaches infinity, the series
has no sum.
Example # 3
• Find the sum ofthe series 60+24+9.6+ … .
• Solution:
• A =60 r= .4 Since |r|, 1, S = 60/1-.4 =100
1
Example # 4
•
A rubber ball is dropped to the floor from a height of 27 feet. If the ball rises at
each rebound to a height 2/3 its previous height, find the total distance the ball
travels before coming to rest.
•
27
18
18
12 and so forth
floor
Notice it must include the ball going up and then back down.
27 + 2(18)+ 2(12)… .
S = 27
= 81 but (except for the 27) each m easure
1-2/3
appears twice 81-27=54X2=108 +27
=135
HW # 42
•Section 12-3
•Pp. 781-783
•#15-29 odds, 30,45,47