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p.460. 7 .4 Find Sums of Infinite Geometric 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?. 1. 1. 1. Consider the infinite geometric series. 1. +. +.

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p.460

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

  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?

  3. 1 1 1 Consider the infinite geometric series 1 + + + 16 8 4 2 1 + + . . . . Find and graph the partial sums Snfor n = 1, 2, 3, 4, and 5. Then describe what happens to Snas n increases. 32 S1 = = 0.5 + = 0.75 S2 = 1 1 2 4 0.88 1 S3 = + + 2 1 1 1 1 1 1 1 1 1 1 + + 0.94 S4= + 4 2 8 4 2 2 8 4 8 16 1 1 + + S5 = + 0.97 + 16 32 SOLUTION From the graph, Snappears to approach 1as nincreases.

  4. The sum of an infinite geometric series

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

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

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

  8. s5 + + + + = • Consider the series + + + + + . . . . Find and graph the partial sumsSnforn = 1, 2, 3, 4 and 5. Then describe what happens to Snas nincreases. 2 S4 + + + = 5 2 4 4 4 4 4 2 2 2 2 S1 0.4 25 25 25 25 25 5 5 5 5 5 = = 8 8 8 8 16 16 16 125 125 125 125 625 625 625 14 S2 0.56 32 32 + = = = 25 3125 3125 Find the sums of the infinite geometric series. SOLUTION S3  0.62 + + = 0.4 + 0.16 + 0.64 =  0.65 = 0.62 + 0.0256 = 0.65 + 0.01024  0.662

  9. ANSWER Snappearsto be approaching ⅔ asnincreases. Graph it…

  10. 8 n – 1 3. 3 n = 1 5 For this series, a1 = 3 andr = 4 = 5 4 a1 The sum formula does not apply whenr ≥ 1 S = 1 – r Does not exist. It has no sum. ANSWER Find the sum of the infinite geometric series, if it exists. SOLUTION

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

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

  13. 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

  14. The repeating decimal 0.242424. . . is as a fraction. a1 = 1 – r 24(0.01) 0.24 24 = 99 0.99 1 – 0.01 = = 8 8 ANSWER 33 33 = Write 0.242424. . . as a fraction in lowest terms. 0.242424. . . = 24(0.01) + 24(0.01)2 + 24(0.01)3 + · · · 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.

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

  16. 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.

  17. 7.4 Assignment: p. 463 3-31 odd

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