p.460

7.4 Find Sums of Infinite 7.4 Find Sums of Infinite Geometric SeriesGeometric Series

What is the formula for finding the sum of an infinite geometric series?

Does an infinite geometric series have a sum if the

How do you write a

repeating decimal as a fraction?

?1r

SOLUTION

S1 =12 = 0.5

S2 =12

14+ = 0.75

18S3 =

12

14+ + 0.88

. . . . Find and graph the partial sums Sn for n = 1, 2, 3, 4, and 5. Then describe what happens to Sn as n increases.

Consider the infinite geometric series 12

14

18

+ + 116+

132+ +

S4=12

14+

18+ 1

16+ 0.94

S5 =12

14+

18+ 1

16+ 132+ 0.97

From the graph, Sn appears to approach 1 as n increases.

The sum of an The sum of an infinite geometric infinite geometric

seriesseries

1r if , 1

1

r

aS

sum. no is there,1 If r

ExampleExample: Find the sum of the : Find the sum of the infinite geometric series.infinite geometric series.

1

1)1.0(2i

i

For this series, aFor this series, a11=2 & r=0.1=2 & r=0.1

1.1

2

S

9

20

9.

2

Find the sum of the infinite geometric series.

a.

5(0.8)i – 18

i = 1

SOLUTION

a. For this series, a1 = 5 and r = 0.8.

S =a1

1 – r = 1 – 0.85

= 25

Find the sum of the infinite geometric series.

SOLUTION

34

916

2764

b. + – +. . .1 –

S =a1

1 – r =1

( )1 – 34

= 47

b. For this series, a1 = 1 and r = – . 34

Find the sums of the infinite geometric series.

SOLUTION

S1 0.4 =25 =

S2 0.56= 25 + 4

25 = 1425 =

1. Consider the series + + + + + . . . . Find and graph the partial sums Sn for n = 1, 2, 3, 4 and 5. Then describe what happens to Sn as n increases.

25

425 125

862516

312532

1258S3 = 2

5 + 425 + = 0.4 + 0.16 + 0.64 0.62

= 0.62 + 0.0256 0.6562516

1258S4 = 2

5 + 425 + +

= 0.65 + 0.01024 0.662

1258

312532s5 = 2

5 + 425 + 625

16+ +

Sn appears to be approaching ⅔

as n increases.

ANSWER

Find the sum of the infinite geometric series, if it exists.

For this series, a1 = 3 and r =54

S =a1

1 – r =

3. 8

n = 1

n – 1543

The sum formula does not apply when r ≥ 1

SOLUTION

Does not exist. It has no sum.ANSWER

ExampleExample: Find the sum of the : Find the sum of the series:series: ...

9

4

3

4412 So, a1=12

and r=1/3

31

1

12

S

32

12S

2

36S

S=18

Pendulums A pendulum that is released to swing freely travels 18 inches on the first swing. On each successive swing, the pendulum travels 80% of the distance of the previous swing. What is the total distance the pendulum swings?

The total distance traveled by the pendulum is:

d = 18 + 18(0.8) + 18(0.8)2 + 18(0.8)3 + · · ·

a1

1 – r=

= 90

181 – 0.8=

Write formula for sum.

Substitute 18 for a1 and 0.8 for r.

Simplify.The pendulum travels a total distance of 90 inches, or 7.5 feet.

SOLUTION

Example: An infinite geom. Series has aExample: An infinite geom. Series has a11=4 & =4 &

a sum of 10. What is the common ratio?a sum of 10. What is the common ratio?

r

aSuse

1 1

r

1

410

10(1-r)=4

1-r = 2/5

-r = -3/5

5

3r

Write 0.242424. . . as a fraction in lowest terms.

0.242424. . . = 24(0.01) + 24(0.01)2 + 24(0.01)3 + · · ·a1

1 – r=

24(0.01)1 – 0.01=

0.240.99

=

2499=

833=

Write formula for sum.

Substitute 24(0.01) for a1 and 0.01 for r.

Simplify.

Write as a quotient of integers.

Reduce fraction to lowest terms.

The repeating decimal 0.242424. . . is833 as a fraction.

ANSWER

Example: Write 0.181818… as a Example: Write 0.181818… as a fraction.fraction.

0.181818…=18(.01)+18(.01)0.181818…=18(.01)+18(.01)22+18(.01)+18(.01)33+…+…

Now use the rule for the sum!Now use the rule for the sum!

r

a

11

01.1

18.

99.

18.

11

2

What is the formula for find the sum of an infinite geometric series?

Does an infinite geometric series have a sum if the

No!How do you write a repeating decimal as a fraction?Use the rule for sum and substitute in for

a1 and r.

?1r

1r if , 1

1

r

aS

r

aS

1 1

7.4 Assignment:7.4 Assignment:

p. 463p. 463

3-31 odd3-31 odd