• 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?

Consider the following sequence: 16, 8, 4, ….

Notes 12-3 and 12-5: Infinite Sequences

and Series, Summation Notation

I. Infinite Sequences and Series

A. Concept and Formula

• 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:

1

1n

aS

r

Notice that 𝑟 < 1. If 𝑟 > 1, the sum does not exist.

The series must also be geometric. Why?

Ex 1: Find the sum of the series 21 − 3 +3

7−⋯

𝑆𝑛 =𝑎11 − 𝑟

𝑆𝑛 =21

1 − (−17)

𝑟 =𝑎2

𝑎1=−3

21=−1

7

𝑆𝑛 = 18.375

Ex 2: Find the sum of the series 60 + 24 + 9.6…

𝑆𝑛 =𝑎11 − 𝑟

𝑆𝑛 =60

1 − .4

𝑟 =𝑎2𝑎1=24

60= .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 completerevolutions will the flywheel make before coming to a stop?

𝑆𝑛 =𝑎11 − 𝑟

𝑆𝑛 =170

1 −25

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+

762

1,000,000+

762

1,000,000,000+⋯

In this series, a1=762

1000and r =

1

1000

𝑆𝑛 =𝑎11 − 𝑟=

7621000

1 −11000

=762

999=254

333

Ex 2: Write 0.123123123… as a fraction using an Infinite Geometric Series.

0. 123 =123

1000+

123

1,000,000+

123

1,000,000,000+⋯

In this series, a1=123

1000and r =

1

1000

𝑆𝑛 =𝑎11 − 𝑟=

1231000

1 −11000

=123

999

Ex 3: Show that 12.33333… = 121

3using a geometric series.

First, write the repeating part as a fraction.

0. 7 =3

10+3

100+3

1,000+⋯

In this series, a1=3

10and r =

3

10

𝑆𝑛 =𝑎11 − 𝑟=

310

1 −110

=3

9=1

3

So, 12.33333… = 121

3

II. Sigma Notation/Summation NotationIn 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.

Summation

symbol

Upper Limit (greatest value of n)

Lower Limit (least value of n)

Expression for

the general

term

Substitute n = 1 into the equation and continue through n = 3.

(5*1 + 1) + (5*2 +1) + (5*3 + 1) =

6+11+16

33

This is read “the summation from n = 1 to 3 of 5n + 1”.

Expanded

Form

A. Writing in Expanded Form and Evaluating

Ex 1: Write the following in expanded form and evaluate:

N = 10

1st = 1

Last = 10

a1= 1 - 3 = -2

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:

25)5(5)72(2

10nS

)(2

1 nn aan

S

Expanded form:

-2 + (-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7

Evaluate: 25

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

2−64

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) 24𝑛

The denominator has factorials, so it should be n!

So we have

𝑛=1

4(−1)24𝑛

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.

𝑆𝑛 =𝑎1 − 𝑎1𝑟

𝑛

1 − 𝑟

𝑆𝑛 =1−1(2)9

1−2

𝑆𝑛 =1−512

1−2

𝑆𝑛 = 511

The loser of all nine holes would have to pay $511.

• c.

•

𝑛=1

9

2𝑛−1

9 total holes

Start at the first hole

Each time is

doubled, we

start with 1$, so

20 = 1, 21 = 2,

22 = 4, etc.