Notes 12-3 and 12-5: Infinite Sequences
and Series, Summation Notation
I. Infinite Sequences and Series
A. Concept and Formula
Consider the following sequence: 16, 8, 4, ….
•
•
•
•
What kind of sequence is it?
Find the 18th term.
Now find the 20th, 25th, and 50th.
So …the larger n is the more the
sequence approaches what?
• W=When 𝑟 < 1, as n increases, the terms of the
sequence will decrease, and ultimately approach zero.
Zero is the limit of the terms in this sequence.
• What will happen to the Sum of the Series?
It will reach a limit as well.
The sum, Sn, of an infinite
geometric series for
which 𝑟 < 1 is given by
the following formula:
a1
Sn 
1 r
Notice that 𝑟 < 1. If 𝑟 > 1, the sum does not exist.
The series must also be geometric. Why?
3
7
Ex 1: Find the sum of the series 21 − 3 + − ⋯
𝑟=
𝑎2
𝑎1
=
−3
21
=
𝑎1
𝑆𝑛 =
1−𝑟
𝑆𝑛 =
21
1
1 − (− )
7
𝑆𝑛 = 18.375
−1
7
Ex 2: Find the sum of the series 60 + 24 + 9.6 …
𝑎2 24
𝑟=
=
= .4
𝑎1 60
𝑎1
𝑆𝑛 =
1−𝑟
60
𝑆𝑛 =
1 − .4
𝑆𝑛 = 100
B. Applications
Ex 1: Francisco designs a toy with a
rotary flywheel that rotates at a
maximum speed of 170 revolutions per
minute. Suppose the flywheel is
operating at its maximum speed for one
minute and then the power supply to the
toy is turned off. Each subsequent
minute thereafter, the flywheel rotates
two-fifths as many times as in the
preceding minute. How many complete
revolutions will the flywheel make before
coming to a stop?
𝑎1
𝑆𝑛 =
1−𝑟
170
𝑆𝑛 =
2
1−
5
𝑆𝑛 = 283.3333
It makes 283 complete
revolutions before it stops.
Ex 2: A tennis ball dropped from a height of
24 feet bounces .75% of the height from
which it fell on each bounce. What is the
vertical distance it travels before coming to
rest?
C. Writing Repeating Decimals as Fractions
• To write a repeating decimal as a fraction, start by writing it as an
infinite geometric sequence.
Ex. 1: Write 0. 762 as a fraction
0. 762 =
762
1000
+
In this series,
762
762
+
1,000,000 1,000,000,000
762
a1=
1000
and r =
1
1000
762
𝑎1
762 254
1000
𝑆𝑛 =
=
=
=
1
1−𝑟 1−
999 333
1000
+⋯
Ex 2: Write 0.123123123… as a fraction using an Infinite Geometric
Series.
0. 123 =
123
1000
In this series,
+
123
123
+
1,000,000 1,000,000,000
123
a1=
1000
and r =
1
1000
123
𝑎1
123
1000
𝑆𝑛 =
=
=
1−𝑟 1− 1
999
1000
+⋯
Ex 3: Show that 12.33333… =
1
12
3
using a geometric series.
First, write the repeating part as a fraction.
0. 7 =
3
10
3
3
+ +
100 1,000
In this series,
3
a 1=
10
+⋯
and r =
3
10
3
𝑎1
3 1
10
𝑆𝑛 =
=
= =
1−𝑟 1− 1
9 3
10
So, 12.33333… =
1
12
3
II. Sigma Notation/Summation Notation
In mathematics, the uppercase Greek letter sigma is often used to indicate a sum or series. This is called
sigma notation. The variable n used with sigma notation is called the index of summation.
Upper Limit (greatest value of n)
Expression for
the general
term
Summation
symbol
Lower Limit (least value of n)
This is read “the summation from n = 1 to 3 of 5n + 1”.
Substitute n = 1 into the equation and continue through n = 3.
(5*1 + 1) + (5*2 +1) + (5*3 + 1) =
6+11+16
33
Expanded
Form
A. Writing in Expanded Form and Evaluating
Ex 1: Write the following in expanded form and evaluate:
N = 10
1st = 1
Expanded form:
-2 + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7
Last = 10
a1= 1 - 3 = -2
Evaluate: 25
a10 = 10 - 3 = 7
Notice that since this is an arithmetic series, we could also use the formula for a finite
arithmetic series to evaluate:
n
Sn  (a1  an )
2
Sn 
10
( 2  7)  5(5)  25
2
Ex 2: Find the number of terms, the first term and the last term. Then
evaluate the series:
N=4
1st = 2
Last = 5
Expanded for:
4+9+16+25
Evaluate: 54
Note: this is NOT an
arithmetic series. You can
NOT use the formula; you
have to manually crunch out
all the values.
B. n Factorial
• As you have seen, not all
sequences are arithmetic or
geometric. Some important
sequences are generated by
products of consecutive
integers. The product n(n – 2) …
3 * 2 * 1 is called n factorial and
is symbolized n!.
• As a rule, 0! = 1
• The table at right shows just
how quickly the numbers can
grown. Copy down the first 7
rows. You will need to recognize
this pattern in a subsequent
example.
C. Writing a series in sigma notation
• Ex 1: 102 + 104 + 106 + 108 + 110 + 112
n = 6 terms
1st term = 1
Rule: Hmmmm. . . .
Rule = 100 + 2n
• Ex 2:Write in sigma notation:
4
−
1
16
64
+ −
2
6
256
+
24
n = 4 terms
1st term = 1
Rule: Hmmmm. . . .
The numerator has powers of 4, and the signs rotate back and forth between positive
and negative. The means the numerator should be (−1) 2 4𝑛
The denominator has factorials, so it should be n!
So we have
4
𝑛=1
(−1)2 4𝑛
𝑛!
D. Applications
• Ex 1: During a nine-hole charity golf
match, one player presents the
following proposition: The loser of
the first hole will pay $1 to charity,
and each succeeding hole will be
worth twice as much as the hole
immediately preceding it.
• a. How much would a losing player
pay on the 4th hole?
• b. How much would a player lose if
he or she lost all nine holes?
• c. Represent the sum using sigma
notation.
• a. Since the sequence is
geometric, we can use the
formula for the nth term of a
geometric sequence.
an = a1rn–1
an = 1(2)4–1
an = (2)3
an = 8
The loser would have to pay $8.
• b. We can use the formula for a finite
geometric series.
𝑎1 − 𝑎1 𝑟 𝑛
𝑆𝑛 =
1−𝑟
𝑆𝑛 =
• c.
•
9 total holes
1−1(2)9
1−2
9
1−512
1−2
𝑛=1
𝑆𝑛 =
𝑆𝑛 = 511
The loser of all nine holes would have to
pay $511.
Each time is
doubled, we
start with 1$, so
20 = 1, 21 = 2,
22 = 4, etc.
2𝑛−1
Start at the first hole