10.3 Geometric and Harmonic Series Contemporary Calculus 1 10.3 GEOMETRIC AND HARMONIC SERIES This section uses ideas from Section 10.2 about series and their convergence to investigate some special types of series. Geometric series are very important and appear in a variety of applications. Much of the early work in the 17th century with series focused on geometric series and generalized them. Many of the ideas used later in this chapter originated with geometric series. It is easy to determine whether a geometric series converges or diverges, and when one does converge, we can easily find its sum. The harmonic series is important as an example of a divergent series whose terms approach zero. A final type of series, called "telescoping," is discussed briefly. Telescoping series are relatively uncommon, but their partial sums exhibit a particularly nice pattern. Geometric Series: ∑k=0∞ C.r k = C + C.r + C.r2 + C.r3 + . . . Example 1: Bouncing Ball: A "super ball" is thrown 10 feet straight up into the air. On each bounce, it rebounds to four fifths of its previous height (Fig. 1) so the sequence of heights is 10 feet, 8 feet, 32/5 feet, 128/25 feet, etc. (a) How far does the ball travel (up and down) during its nth bounce? (b) Use a sum to represent the total distance traveled by the ball. Solution: Since the ball travels up and down on each bounce, the distance traveled during each bounce is twice the height of the ball on that bounce so d1 = 2(10 feet) = 20 feet, d2 = 16 feet, d3 = 64/5 feet, and, in general, dn = 45 .dn–1 . Looking at these values in another way, d1 = 20 , d2 = 45 .(20) , d3 = 45 d2 = 45 .45 .20 = ( 45 )2 (20) , d4 = 45 .d3 = 45 .( ( 45 )2 .(20) ) = ( 45 )3 .(20) , and, in general, dn = ( 45 )n–1 .(20) . In theory, the ball bounces up and down forever, and the total distance traveled by the ball is the sum of the distances traveled during each bounce (an up and down flight): (first bounce) + (second bounce) + (third bounce) + (forth bounce) + . . . = 20 + 45 (20) + ( 45 )2 (20) + ( 45 )3 (20) + . . . = 20.( 1 + 45 + ( 45 )2 + ( 45 )3 + . . . ) = 20. ∑k=0∞ ( 45 ) k .10.3 Geometric and Harmonic Series Contemporary Calculus 2 Practice 1: Cake: Three calculus students want to share a small square cake equally, but they go about it in a rather strange way. First they cut the cake into 4 equal square pieces, each person takes one square, and one square is left (Fig. 2). Then they cut the leftover piece into 4 equal square pieces, each person takes one square and one square is left. And they keep repeating this process. (a) What fraction of the total cake does each person "eventually" get? (b) Represent the amount of cake each person gets as a geometric series: (amount of first piece) + (amount of second piece) + . . . Each series in the previous Example and Practice problems is a Geometric series, a series in which each term is a fixed multiple of the previous term. Geometric series have the form ∑k=0∞ C.r k = C + C.r + C.r2 + C.r3 + . . . = C. ∑k=0∞ r k with C ≠ 0 and r ≠ 0 representing fixed numbers. Each term in the series is r times the previous term. Geometric series are among the most common and easiest series we will encounter. A simple test determines whether a geometric series converges, and we can even determine the "sum" of the geometric series. Geometric Series Theorem The geometric series ∑k=0∞ r k = 1 + r + r2 + r3 + ...  converges to 11 – rif | r | < 1 diverges if | r | ≥ 1 Proof: If | r | ≥ 1, then | rk | approaches 1 or +∞ as k becomes arbitrarily large, so the terms ak = rk of the geometric series do not approach 0. Therefore, by the nth term test for divergence, the series diverges. If | r | < 1, then the terms ak = rk of the geometric series approach 0 so the series may or may not converge, and we need to examine the limit of the partial sums sn = 1 + r + r2 + r3 + . . . + rn of the series. For a geometric series, a clever insight allows us to calculate those partial sums:10.3 Geometric and Harmonic Series Contemporary Calculus 3 (1 – r).sn = (1 – r).(1 + r + r2 + r3 + . . . + rn) = 1.(1 + r + r2 + r3 + . . . + rn) – r.(1 + r + r2 + r3 + . . . + rn) = (1 + r + r2 + r3 + . . . + rn) – ( r + r2 + r3 + r4 + . . . + rn + rn+1 ) = 1 – rn+1 . Since | r | < 1 we know r ≠ 1 so we can divide the previous result by 1 – r to get sn = 1 + r + r2 + r3 + . . . + rn = 1 – rn+11 – r = 11 – r – rn+11 – r . This formula for the nth partial sum of a geometric series is sometimes useful, but now we are interested in the limit of sn as n approaches infinity. Since | r | < 1, rn+1 approaches 0 as n approaches infinity, so we can conclude that the partial sums sn = 11 – r – rn+11 – r approach ! 11" r (as "n # $"). The geometric series ∑k=0∞ r k converges to the value 11 – r when –1 < r < 1 . Finally, ∑k=0∞ C.r k = C. ∑k=0∞ r k so we can easily determine whether or not ∑k=0∞ C.r k converges and to what number. Example 2: How far did the ball in Example 1 travel? Solution: The distance traveled, 20( 1 + 45 + ( 45 )2 + ( 45 )3 + . . . ) , is a geometric series with C = 20 and r = 4/5. Since | r | < 1, the series 1 + 45 + ( 45 )2 + ( 45 )3 + . . . converges to 11 – r = 11 – 4/5 = 5, so the total distance traveled is 20( 1 + 45 + ( 45 )2 + ( 45 )3 + . . . ) = 20( 5 ) = 100 feet. Repeating decimal numbers are really geometric series in disguise, and we can use the Geometric Series Theorem to represent the exact value of the sum as a fraction. Example 3: Represent …