Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
INFINITE SERIES
So far, we have considered only finite series. Now it's time to head off to infinity. Under what

circumstances will
a
k
converge? That is, when will the sum of an infinite number of terms
k 1
still be finite? In this course, we will only begin to answer the question.

1.
a. Consider
 (3k  1) .
What happens to this sum with more and more terms? Why?
k 1

b. Consider
 (4  0.3 j) . What is the sum?
Why?
j 1
c. What may be said about the sum of any infinite arithmetic series?
2.
Given the geometric series: S = 12  14  81  161  321 L , complete the table for the sequence
of partial sums, then plot the points.
n
1
Sn
2
1
4
1
2
1
2
3
Sn
1
4
4
5
6
1
6
n
Find the horizontal asymptote that S n seems to be approaching. Graph and label it.
Seq & Ser. 8.1
Rev. S06
Mathematical Investigations IV
Name
3.
Let’s re-examine our formula for the sum of a geometric series:
a  a  rn
Sn 
with first term a and common ratio r.
1 r
We need to examine this formula as n gets larger and larger. This becomes a question
about the size of r n .
a.
First choose an r such that | r | < 1. For example, let r  1 :
As n grows larger:

1
3
n
3
Try another. Let r = ________.

As n grows larger: r
n
 
As n grows larger,  13
n
 
As n grows larger, r
n
In general, if | r | < 1, then r n 
In terms of r, what happens to the formula for Sn?
b.
Next choose r such that | r | > 1. For example, let r  4 .
As n grows larger:
3

4
3
n
Try another. Let r = ________.

As n grows larger: r
n
 
As n grows larger, r
n
What happens to the formula for Sn?
c.
Consider: r = 1 or r = –1
As n grows larger,
1 
n
 
n
and 1 
If r = 1, what does the series become? What happens to the sum as n grows larger?
d.
Summarizing, under what conditions will the sum Sn be finite as n grows larger?
Seq & Ser. 8.2
Rev. S06
Mathematical Investigations IV
In short, if | r | < 1, then Sn 
Name
a
as n grows larger. For such r, we say that the infinite
1 r
a
. However, if | r | > 1, the sum of an infinite geometric series
1 r
"blows up" or "diverges."
series converges to S 
4.
Find the sum of each infinite geometric series, if it has a finite sum. If not, why not?

a.
 9 
k 1
2
3
k
List several terms first.
b. 12 – 9 +
27
4
– ...
c. 4  6  9  27
2 L
d.
Find S = 12  14  81  161  321 L . Does this agree with your asymptote from problem 2
on sheet 8.1?
5.
A ball is dropped from a height of 9 meters and rebounds 78 ths of its height on each
bounce. How far will the ball travel before coming to rest?
6.
Find the sum as a fraction:
0.7 + .07 + .007 + .0007 + ...
7.
Write .212121... as an infinite series and
then as a fraction.
Seq & Ser. 8.3
Rev. S06
Mathematical Investigations IV
Name
Telescoping Series:
Sometimes an infinite telescoping series converges. Determine whether the following converge or
diverge. If it converges, find its value.
 6
6 
8.  

 2k  1 2k  3
k 1

9.
 log
k 1

10.
 k  1
 k  2 
k 1 
, 4  (intervals)
k


U
k 1

11.
2
k
k 3
(Note: This is an infinite product, so all terms are multiplied, rather than added.)
k 1
Seq & Ser. 8.4
Rev. S06