1. 7.4 Find Sums of Infinite Geometric Series
p.460

2. 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?

4. Consider the infinite geometric series 1 2 4 8 + 16
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 1 2 4 8 + 16 32
SOLUTION
S1 = 1 2
= 0.5
S2 = 1 2 4 +
= 0.75 1 8
S3 = 2 4 +
0.88
From the graph, Sn appears to approach 1 as n increases.
S4= 1 2 4 + 8 16
0.94
S5 = 1 2 4 + 8 16 32
0.97

6. The sum of an infinite geometric series

7. Example: Find the sum of the infinite geometric series.
For this series, a1=2 & r=0.1

8. Find the sum of the infinite geometric series.a.
5(0.8)i – 1 8
i = 1
SOLUTION
a. For this series, a1 = 5 and r = 0.8.
S = a1
1 – r =
1 – 0.8 5
= 25

9. ( ) Find the sum of the infinite geometric series. 3 9 27 b. . . . 1 –4 9 16 27 64 b. + –
. . .
1 –
SOLUTION
b. For this series, a1 = 1 and r = – 3 4
S = a1
1 – r = 1
( )
1 – 3 4 7

10. Find the sums of the infinite geometric series.
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. 2 5 4 25
125 8
625 16
3125 32
SOLUTION S = 2 5 S = 2 5 + 4 25 14 2 5 4 25
125 8 S3 = + + =
 0.62
625 16
125 8 S4 = 2 5 + 4 25 =
 0.65
125 8
3125 32 s5 = 2 5 + 4 25
625 16 = 

11. Graph it…
Sn appears to be approaching ⅔ as n increases.
ANSWER

12. Find the sum of the infinite geometric series, if it exists.3. 8
n = 1
n – 1 5 4 3
SOLUTION
For this series, a1 = 3 and r = 5 4
S = a1
1 – r =
The sum formula does not apply when r ≥ 1
Does not exist.
It has no sum.
ANSWER

13. Example: Find the sum of the series:
So, a1=12 and r=1/3
S=18

14. The total distance traveled by the pendulum is:
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?
SOLUTION
The total distance traveled by the pendulum is:
d = (0.8) + 18(0.8)2 + 18(0.8)3 + · · · a1
1 – r =
Write formula for sum. 18
1 – 0.8 =
Substitute 18 for a1 and 0.8 for r.
The pendulum travels a total distance of 90 inches, or 7.5 feet.
= 90
Simplify.

15. Example: An infinite geom. Series has a1=4 & a sum of 10
Example: An infinite geom. Series has a1=4 & a sum of 10. What is the common ratio?
10(1-r)=4
1-r = 2/5
-r = -3/5

16. Write 0.242424. . . as a fraction in lowest terms.
= 24(0.01) + 24(0.01)2 + 24(0.01)3 + · · · a1
1 – r =
Write formula for sum.
24(0.01)
1 – 0.01 =
Substitute 24(0.01) for a1 and 0.01 for r.
0.24
0.99 =
Simplify. 24 99 =
Write as a quotient of integers. 8 33 =
Reduce fraction to lowest terms.
The repeating decimal is 8 33
as a fraction.
ANSWER

17. Example: Write 0.181818… as a fraction.
…=18(.01)+18(.01)2+18(.01)3+…
Now use the rule for the sum!

18. 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.

19. 7.4 Assignment:
p. 463
3-31 odd