10.2 Infinite Series Contemporary Calculus 1 10.2 INFINITE SERIES Our goal in this section is to add together the numbers in a sequence. Since it would take a "very long time" to add together the infinite number of numbers, we first consider finite sums, look for patterns in these finite sums, and take limits as more and more numbers are included in the finite sums. What does it mean to add together an infinite number of terms? We will define that concept carefully in this section. Secondly, is the sum of all the terms a finite number? In the next few sections we will examine a variety of techniques for determining whether an infinite sum is finite. Finally, if we know the sum is finite, can we determine the value of the sum? The difficulty of finding the exact value of the sum varies from very easy to very, very difficult. Example 1: A golf ball is thrown 9 feet straight up into the air, and on each bounce it rebounds to two thirds of its previous height (Fig. 1). Find a sequence whose terms give the distances the ball travels during each successive bounce. Represent the total distance traveled by the ball as a sum. Solution: The heights of the successive bounces are 9 feet, ( 23 ).9 feet, ( 23 ).[( 23 ).9] feet, ( 23 )3 .9 feet, and so forth. On each bounce, the ball rises and falls so the distance traveled is twice the height of that bounce: 18 feet, ( 23 ).18 feet, ( 23 ).( 23 ).18 feet, ( 23 )3 .18 feet , ( 23 )4 .18 feet , . . . . The total distance traveled is the sum of the bounce–distances: total distance = 18 + ( 23 ).18 + ( 23 ).( 23 ).18 + ( 23 )3 .18 + ( 23 )4 .18+ . . . = 18 { 1 + 23 + ( 23 )2 + ( 23 )3 + ( 23 )4 + . . . } At the completion of the first bounce the ball has traveled 18 feet. After the second bounce, it has traveled 30 feet, a total of 38 feet after the third bounce, 43 13 feet after the fourth, and so on. With a calculator and some patience, we see that after the 20th bounce the ball has traveled a total of approximately 53.996 feet, after the 30th bounce approximately 53.99994 feet, and after the 40th bounce approximately 53.9999989 feet.10.2 Infinite Series Contemporary Calculus 2 Practice 1: A tennis ball is thrown 10 feet straight up into the air, and on each bounce it rebounds to 40% of its previous height. Represent the total distance traveled by the ball as a sum, and find the total distance traveled by the ball after the completion of its third bounce. (Fig. 2) Infinite Series The infinite sums in the Example and Practice are called infinite series, and they are the objects we will start to examine in this section. Definitions An infinite series is an expression of the form a1 + a2 + a3 + a4 + . . . or ∑k=1∞ ak . The numbers a1, a2, a3, a4, . . . are called the terms of the series. (Fig. 3) Example 2: Represent the following series using the sigma notation. (a) 1 + 1/3 + 1/9 + 1/27 + . . . , (b) –1 + 1/2 – 1/3 + 1/4 – 1/5 + . . . , (c) 18( 2/3 + 4/9 + 8/27 + 16/81 + . . . ) (d) 0.777 ... = 7/10 + 7 /100 + 7/1000 + ... , and (e) 0.222... Solution: (a) 1 + 1/3 + 1/9 + 1/27 + . . . = ∑k=0∞ ( 13 ) k or ∑k=1∞ ( 13 ) k–1 (b) –1 + 1/2 – 1/3 + 1/4 – 1/5 + . . . = ∑k=1∞ (–1) k 1k (c) 18 ∑k=1∞ ( 23 ) k (d) 0.777 ... = 7/10 + 7 /100 + 7/1000 + ... = ∑k=1∞ 710k (e) ∑k=1∞ 210k Practice 2: Represent the following series using the sigma notation. (a) 1 + 2 + 3 + 4 + . . . , (b) –1 + 1 – 1 + 1 – . . . (c) 2 + 1 + 1/2 + 1/4 + . . . (d) 1/2 + 1/4 + 1/6 + 1/8 + 1/10 +. . . (e) 0.111...10.2 Infinite Series Contemporary Calculus 3 In order to determine if the infinite series adds up to a finite value, we examine the sums as more and more terms are added. Definition The partial sums sn of the infinite series ∑k=1∞ ak are the numbers s1 = a1, s2 = a1 + a2, s3 = a1 + a2 + a3 , . . . In general, sn = a1 + a2 + a3 + . . . + an = ∑k=1n ak or , recursively, as sn = sn–1 + an . The partial sums form the sequence of partial sums { sn } . Example 3: Calculate the first 4 partial sums for the following series. (a) 1 + 1/2 + 1/4 + 1/8 + 1/16 + . . . , (b) ∑k=1∞ (–1) k , and (c) ∑n=1∞ 1n . Solution: (a) s1 = 1, s2 = 1 + 1/2 = 3/2, s3 = 1 + 1/2 + 1/4 = 7/4, s4 = 1 + 1/2 + 1/4 + 1/8 = 15/8 It is usually easier to use the recursive version of sn : s3 = s2 + a3 = 3/2 + 1/4 = 7/4 and s4 = s3 + a4 = 7/4 + 1/8 = 15/8. (b) s1 = (–1)1 = –1, s2 = s1 + a2 = –1 + (–1)2 = 0, s3 = s2 + a3 = 0 + (–1)3 = –1, s4 = 0. (c) s1 = 1, s2 = 3/2, s3 = 11/6, s4 = 25/12 . Practice 3: Calculate the first 4 partial sums for the following series. (a) 1 – 1/2 + 1/4 – 1/8 + 1/16 – . . . , (b) ∑k=1∞ ( 13 ) k , and (c) ∑n=2∞ (–1)nn . If we know the values of the partial sums sn , we can recover the values of the terms an used to build the sn. Example 4: Suppose s1 = 2.1 , s2 = 2.6 , s3 = 2.84 , and s4 = 2.87 are the first partial sums of ∑k=1∞ ak . Find the values of the first four terms of { an } . Solution: s1 = a1 so a1 = 2.1 . s2 = a1 + a2 so 2.6 = 2.1 + a2 and a2 = 0.5 . Similarly, s3 = a1 + a2 + a3 so 2.84 = 2.1 + 0.5 + a3 and a3 = 0.24. Finally, a4 = 0.03 .10.2 Infinite Series Contemporary Calculus 4 An alternate solution method starts with a1 = s1 and then uses the fact that sn = sn–1 + an so an = sn – sn–1 . Then a2 = s2 – s1 = 2.6 – 2.1 = 0.5 …