880 CHAPTER 12 Sequences; Induction; the Binomial Theorem Determining Whether a Geometric Series Converges or Diverges Determine whether the geometric series 2 2 3 2 4 3 8 9 k k 1 1 ∑ ( ) = + + + = ∞ − converges or diverges. If it converges, find its sum. EXAMPLE 7 Solution Comparing 2 2 3 k k 1 1 ∑ ( ) = ∞ − to a r , k k 1 1 1 ∑ = ∞ − the first term is a 2 1 = and the common ratio is r 2 3 . = Since r 1, < the series converges. Use formula (7) to find its sum: 2 2 3 2 4 3 8 9 2 1 2 3 6 k k 1 1 ∑ ( ) = + + + = − = = ∞ − Repeating Decimals Show that the repeating decimal 0.999 . . . equals 1. EXAMPLE 8 Solution The decimal = + + + = + + + … 0.999 0.9 0.09 0.009 9 10 9 100 9 1000 is an infinite geometric series. Write it in the form a r k k 1 1 1 ∑ = ∞ − and use formula (7). ∑ ∑ ∑ ( ) = + + + = = ⋅ = = ∞ = ∞ − = ∞ − … 0.999 9 10 9 100 9 1000 9 10 9 10 10 9 10 1 10 k k k k k k 1 1 1 1 1 Now Work PROBLEM 53 Comparing this series to a r k k 1 1 1 ∑ = ∞ − we have that a 9 10 1 = and r 1 10 . = Since r 1, < the series converges and its sum is … 0.999 9 10 1 1 10 9 10 9 10 1 = − = = The repeating decimal 0.999 . . . equals 1. Pendulum Swings Initially, a pendulum swings through an arc of length 18 inches. See Figure 16. On each successive swing, the length of the arc is 0.98 of the previous length. (a) What is the length of the arc of the 10th swing? (b) On which swing is the length of the arc first less than 12 inches? (c) After 15 swings, what total distance has the pendulum swung? (d) When it stops, what total distance has the pendulum swung? EXAMPLE 9 Figure 16 18” Solution (a) The length of the first swing is 18 inches. The length of the second swing is 0.98 18 ⋅ inches. The length of the third swing is 0.98 0.98 18 0.98 18 2 ⋅ ⋅ = ⋅ inches. The length of the arc of the 10th swing is 0.98 18 15.007 inches 9 ( ) ⋅ ≈
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