18.2 Infinite Geometric Series 571 PROBLEMS FOR SECTION 18.2 For Problems 1 through 11, determine whether the series converges or diverges. If it converges, find its sum. 1. 1 − 10 + 100 −···+(−10) n +··· 2. 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 +··· 3. 2 3 + 2 9 + 2 27 +···+ 2 3 n +··· 4. 2 3 + 2 + 6 +···+2(3) n +··· 5. 1 − 1 2 + 1 4 − 1 8 + 1 16 +··· 6. 1 4 − 1 8 + 1 16 − 1 32 +··· 7. 2 3 + 1 + 3 2 + 9 4 +··· 8. 3 2 − 3 4 + 3 8 −···+ (−1) n+1 3 2 n +··· 9. e + 1 + e −1 + e −2 + e −3 +··· 10. 2e + 2e 2 + 2e 3 +···+2e n +··· 11. (2e) −2 + (2e) −3 + (2e) −4 +···+(2e) −n +··· 12. Find the sum of the following. (If there is no finite sum, say so.) (a) 3 + 9 + 27 +···+3 20 (b) 2 3 + 2 3 2 + 2 3 3 +···+ 2 3 n +··· (c) (0.2)(10) + (0.2)(100) + (0.2)(1000) +··· (d) 3 + 3(0.8) + 3(0.8) 2 + 3(0.8) 3 +··· (e) (0.2) + (0.2)(1.3) + (0.2)(1.3) 2 + (0.2)(1.3) 3 +··· (f) 1 + x 2 + x 4 + x 6 +···for −1 <x<1 13. Determine whether each of the following geometric series converges or diverges. If the series converges, determine to what it converges. (a) − 4 3 − 1 2 − 3 16 − 9 128 +··· (b) − 1 100 + 1.1 (100) 2 − 1.21 (100) 3 + 1.331 (100) 4 −··· (c) − 7 10000 + 7 11000 − 7 12100 + 7 13310 −··· (d) 1 − x + x 2 − x 3 +···for |x| < 1 14. Write each of the following series first as a repeating decimal and then as a fraction. (a) 2 + 2 10 + 2 100 + 2 1000 +··· (b) 3 + 12 10 2 + 12 10 4 + 12 10 6 +··· 572 CHAPTER 18 Geometric Sums, Geometric Series 18.3 A MORE GENERAL DISCUSSION OF INFINITE SERIES In the previous four sections we focused on geometric sums and geometric series. In this section we broaden our discussion to investigate other infinite series. Our focus in this chapter is geometric series, but you will have a better appreciation of geometric series if you have some familiarity with series that are not geometric. Givenaninfinite series a 1 + a 2 + a 3 +···+a n +··· the most basic question to consider is whether the series converges or diverges. Suppose all the terms of the infinite series are positive. Then S n = S(n), the sum of the first n terms, is an increasing function. We know from our study of functions that an increasing function may increase without bound, or it may increase but be bounded, in which case it will be asymptotic to a horizontal line. In the latter case, the function must be increasing at a decreasing rate; in fact, if the function has a horizontal asymptote, its rate of increase must be approaching zero. Similarly, if lim n→∞ S n = L for some finite constant L, then the rate at which S(n) is increasing must be approaching zero. This translates to the observation that if an infinite series is to have any chance at converging, then its terms must be approaching zero, that is, lim n→∞ a n must be 0. partial sums n The partial sums are increasing at a decreasing rate; The partial sums are bounded. partial sums n The partial sums are increasing at a decreasing rate; The partial sums are unbounded. Figure 18.1 If all of the terms of the series are negative, an analogous argument can be made. If some of the terms of an infinite series are positive and others are negative, it is still true that in order for the series to have any shot at converging the terms must be approaching zero. If lim n→∞ a n = k, where k = 0, then the partial sums will be eventually increasing without bound if k>0and eventually decreasing without bound if k<0. 7 Suppose the terms of a series are approaching zero; is this enough to guarantee convergence? In the case of geometric series the answer is “yes”, but in the general case the answer is NO! The situation in general is much more subtle; 8 the next example will convince you of that fact. 7 If lim n→∞ a n does not exist, then the partial sums will be bouncing around and will not converge. 8 We know that a function can be increasing at a decreasing rate and have a horizontal asymptote, but it can also be increasing at a rate tending toward zero and yet be unbounded. Consider, for example, f(x)=ln x. Its rate of increase is given by 1 x . lim n→∞ 1 x = 0, so the rate of increase of ln x tends toward zero as x increases without bound. Nevertheless, as x →∞we know that ln x →∞. 18.3 A More General Discussion of Infinite Series 573 ◆ EXAMPLE 18.7 Does the infinite series 1 2 + 1 3 + 1 4 + 1 5 +···+ 1 n +···converge or diverge? This infinite series is called the harmonic series. SOLUTION The terms of this series are going toward zero: lim n→∞ 1 n = 0. The harmonic series is not a geometric series. Can we put S n into closed form? Let’s look at some partial sums. 1 2 1 2 + 1 3 = 5 6 1 2 + 1 3 + 1 4 = 13 12 1 2 + 1 3 + 1 4 + 1 5 = 77 60 1 2 + 1 3 + 1 4 +···+ 1 n =?? Closed form allowed us to get a firm grip on something rather slippery. In this example our luck has run out. We cannot express the general partial sum in closed form. The expression lim n→∞ 1 2 + 1 3 + 1 4 +···+ 1 n gives us no insight into the convergence or divergence of the series. We need to take a different perspective. We will compare this series to a familiar series. 1 2 = 1 2 1 4 + 1 4 = 1 2 so 1 3 + 1 4 > 1 2 1 8 + 1 8 + 1 8 + 1 8 = 1 2 so 1 5 + 1 6 + 1 7 + 1 8 > 1 2 8 · 1 16 = 1 2 so 1 9 + 1 10 + 1 11 +···+ 1 16 > 1 2 16 · 1 32 = 1 2 so 1 17 + 1 18 + 1 19 +···+ 1 32 > 1 2 and so on Think about this in terms of slices of pies. How many pies must we bake in order to give out slices as dictated by the harmonic series? If the series converges we need only bake some finite number of pies. 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 +···+ 1 16 +···equals 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 +···+ 1 16 +··· half a pie + more than half a pie + more than half a pie + more than half a pie +···, which is greater than or equal to 1 2 + 1 2 + 1 2 + 1 2 + ···. But this latter series diverges; consequently, the same must be true of the harmonic series. By comparing the harmonic 574 CHAPTER 18 Geometric Sums, Geometric Series series with the divergent series 1 2 + 1 2 + 1 2 + 1 2 + ···,wesee that the harmonic series must diverge. (No matter how many pies we bake, we will eventually run out and need more.) ◆ We had previously observed that if an infinite series is to have any chance at converg- ing, then the terms must be going toward zero. Although this is a necessary condition for convergence, a look at the harmonic series shows that it is not enough to guarantee conver- gence; the condition lim n→∞ a n = 0 is necessary for convergence but not sufficient. 9 It is important to realize that if the terms of an infinite series are going to zero, then the series may converge (as is true for all geometric series), yet on the other hand, the series may diverge (as in the example of the harmonic series). A Summary of the Main Principles The Nth Term Test for Divergence: If lim n→∞ a n = 0, then the series a 1 + a 2 + a 3 +···+a n +··· diverges. 10 Warning: This is a test for divergence only! Increasing and Bounded Partial Sums Test: Suppose the terms of a series are all positive. Then S n increases with n. If the partial sums are bounded, that is, there exists a constant M such that S n ≤ M for all n, then it can be shown that lim n→∞ S n exists and is finite. Therefore, the series converges. In many cases the question of convergence or divergence of an infinite series is a very subtle one. Often, for instance, one can determine that a certain series converges without being able to say exactly what it converges to. We will return to infinite series in Chapter 30. Questions of convergence become much simpler if we focus on the special case of the geometric series, and this is our main focus in this chapter. In the case of geometric series, convergence and divergence are straightforward to establish. a + ar + ar 2 +···+ar n +··· converges to a 1−r for |r| < 1 diverges for |r|≥1. In Section 18.4 we introduce some convenient notation for working with series, and in Section 18.5 we apply geometric series to real-world situations. PROBLEMS FOR SECTION 18.3 For Problems 1 through 9, determine whether the series converges or diverges. Explain your reasoning. 1. 1 − 2 + 3 −···+(−1) (n+1)n +··· 2. 1 1000 + 2 1000 + 3 1000 + 4 1000 +··· 3. 2 3 + 2 4 + 2 5 +···+ 2 n +··· 9 “Necessary” versus “sufficient”: In order for a polygon to be called a square it is necessary that it have four sides, but this alone is not sufficient to classify it as a square. In order to win a race it is necessary to finish it, but this alone is not sufficient. 10 Foraninfinite series to diverge it is sufficient that lim n→∞ a n = 0. This, however, is not necessary. To lose a race it is sufficient not to finish it. That, however, is not the only way to lose a race! 18.4 Summation Notation 575 4. 1 2·2 2 + 1 3·2 3 + 1 4·2 4 + 1 5·2 5 +··· (Hint: Compare this term-by-term to a geometric series you know. Choose a convergent geometric series whose terms are larger than the terms of this series.) 5. (sin 1) 2 3 + (sin 2) 2 3 2 + (sin 3) 2 3 3 +···+ (sin n) 2 3 n +··· (Hint: Compare this term-by-term to a geometric series you know. Choose a convergent geometric series whose terms are larger than the terms of this series.) 6. 3 3 3 + 4 4 4 + 5 5 5 +··· 7. −1 2 − 1 3 − 1 4 − 1 5 −···− 1 n −··· 8. 1 2 − 1 2 + 1 2 − 1 2 +··· 9. (1 + 1) 1 + 1 + 1 2 2 + 1 + 1 3 3 +···+ 1+ 1 n n +··· 10. The sum 1 + 2 + 3 + 4 + 5 + ···+n= n k=1 kis not geometric, but we can express it in an easy-to-compute form. Let S = 1 + 2 + 3 + 4 + 5 +···+n. (18.1) Writing the terms from largest to smallest gives S = n + (n − 1) + (n − 2) +···+2+1. (18.2) Add equations (18.1) and (18.2) and divide by two to show that 1 + 2 + 3 + 4 + 5 +···+n= n(n + 1) 2 . 11. Challenge: Use the same line of reasoning as outlined in Problem 10 to show that 1 + 3 + 5 + 7 +···+(2n−1)=n 2 . 12. Give an example of each of the following. (a) An infinite series that converges and whose partial sums are always increasing (b) An infinite series that converges and whose partial sums oscillate around the sum of the series (c) An infinite series that diverges although its terms approach zero (d) An infinite series that diverges but whose partial sums do not grow without bound 18.4 SUMMATION NOTATION Sums whose terms follow a consistent pattern can often be written in a more compact way by using summation notation. Summation notation is only notation; it is a compact shorthand for writing out a sum. We will introduce it via examples. a 1 + a 2 + a 3 +···+a 28 576 CHAPTER 18 Geometric Sums, Geometric Series can be written in shorthand as 28 i=1 a i , where denotes the summation process. 11 The terms of the sum are all of the form a i , where the “i”s form a sequence of consecutive integers starting with the integer indicated at the bottom of and ending with the integer at the top. In this example we successively substitute i = 1, 2, 3, ,28inplace of i in the expression a i and add up these 28 terms. i is called the index. We can choose any letter we like for the index. 28 i=1 a i , 28 j=1 a j , and 28 k=1 a k all are shorthand for a 1 + a 2 + a 3 + ···+a 28 . In fact, 27 q=0 a q+1 is also equivalent to a 1 + a 2 + a 3 + ···+a 28 ; there are infinitely many different ways of putting a sum into summation notation. ◆ EXAMPLE 18.8 i) 14 k=0 2 k is shorthand for 2 0 + 2 1 + 2 2 +···+2 14 . ii) 14 k=0 k 2 is shorthand for 0 2 + 1 2 + 2 2 +···+14 2 . iii) 49 k=3 (−1) k k is shorthand for (−1) 3 3 + (−1) 4 4 + (−1) 5 5 +···+(−1) 49 49. or −3 + 4 − 5 + 6 +···−49. iv) a + ar + ar 2 + ar 3 +···+ar n +··· can be written as ∞ n=0 ar n . v) 1 2 + 1 2 2 + 1 2 3 +···+ 1 2 n +··· can be written as ∞ k=1 1 2 k , or, if we want this to look more explicitly like the general geometric series in part (iv), it can be written ∞ k=0 1 2 1 2 k . vi) 1 2 − 1 2 2 + 1 2 3 +···+ 1 2 33 can be written 32 k=0 1 2 −1 2 k . vii) f 1 n 1 n + f 2 n 1 n + f 3 n 1 n +···+f n n 1 n can be written n i=1 f i n 1 n . NOTE If a sum is geometric, we can begin by identifying “a” (the first term) and “r” (the ratio of any term to the previous one). Write w k=0 ar k and figure out what w ought to be. In part (vi) when we use this approach the upper index is 32 not 33. Writing 33 k=1 −1 − 1 2 k would also be correct. 12 ◆ EXERCISE 18.3 Try these on your own and then check your answers with those given below. (a) Express 2 3 + 2 3 2 + 2 3 3 + ···+ 2 3 40 in summation notation. Do this in two ways; with the index starting at 1 and with the index starting at 0. (b) Express 2 3 2 − 2 3 5 + 2 3 8 − 2 3 11 +···in summation notation. Answers (a) 2 3 + 2 3 2 + 2 3 3 + ···+ 2 3 40 can be written as 40 n=1 2 3 n or as 39 n=0 2 3 2 3 n . This is a geometric sum with a = 2 3 and r = 2 3 . 11 is the uppercase Greek letter sigma; you can think of it as the letter S for sum. It denotes a process, much in the same way that d dx denotes a process or operation. 12 To have the terms of this sum alternate sign, r must be negative. 18.4 Summation Notation 577 (b) This is a geometric series. First identify r by finding the ratio of the second term to the first, r =− 2 3 3 . Once we know r and a,we’re well on our way. The series can be written as ∞ k=0 2 3 2 − 2 3 3k , or, alternatively, ∞ k=2 − 2 3 3k+2 . EXERCISE 18.4 Determine whether each of the following statements is always true or not always true. (a) n i=1 c(1) i = nc (b) n i=1 (a i + b i ) = n i=1 a i + n i=1 b i (c) n i=1 a i b i = a i n i=1 b i (d) n i=1 cb i = c n i=1 b i (e) n i=1 a i b i = n i=1 a i n i=1 b i (f) If a i >b i for i = 1, 2, 3, ,n,then n i=1 a i > n i=1 b i . (Four of the six statements are always true.) ◆ EXAMPLE 18.9 What rational number has the decimal expansion 5.123232323 ? SOLUTION 5.1232323 =5.1 + 0.0232323 =5.1 + 23 10 3 + 23 10 5 + 23 10 7 +···+ 23 10 2n+1 +··· 13 Following 5.1 is a geometric series with a = 23 10 3 and r = 1 10 2 . |r| < 1, so the series converges. We have 5.1 + ∞ n=0 23 10 3 1 100 n = 51 10 + 23 1000 1 − 1 100 = 51 10 + 23 1000 · 100 99 = 51 10 + 23 990 = 5072 990 = 2536 495 . ◆ PROBLEMS FOR SECTION 18.4 For Problems 1 through 10, write the sum using summation notation. 1. 2 3 + 3 3 + 4 3 +···+19 3 2. 2 − 3 + 4 − 5 + 6 −···+100 3. 2 3 + 3 4 + 4 5 +···+100 101 4. (a) 4x 2 + 4x 3 + 4x 4 + 4x 5 +···+ (b) 2x + 3x 2 + 4x 3 + 5x 4 + 6x 5 +···+ 5. 1 − 10 + 100 −···+(−10) n +··· 6. 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 +··· 13 In order to write a general term for this series we need to write 23 10 (odd number) . An even number can be written as 2n for n a positive integer. An odd number can be expressed as 2n + 1or2n−1; hence the general term is given as 23 10 2n+1 . 578 CHAPTER 18 Geometric Sums, Geometric Series 7. (a) 2 3 + 2 9 + 2 27 +···+ 2 3 n +··· (b) 2 3 + 2 + 6 +···+2(3) n +··· 8. (a) 1 − 1 2 + 1 4 − 1 8 + 1 16 +··· (b) 1 4 − 1 8 + 1 16 − 1 32 +··· 9. (a) 2 3 + 1 + 3 2 + 9 4 +··· (b) 3 2 − 3 4 + 3 8 −···+ (−1) n 3 2 n 10. (a) e + 1 + e −1 + e −2 + e −3 +··· (b) 2e + 2e 2 + 2e 3 +···+2e n +··· (c) (2e) −2 + (2e) −3 + (2e) −4 +···+(2e) −n +··· ForProblems 11 through 16, do the following. i. Write out the first two terms of the series. ii. Determine whether or not the series converges. iii. If the series converges, determine its sum. 11. (a) ∞ k=1 (−1) n 3 n (b) ∞ k=2 (−1) n 3 n 12. ∞ n=3 (−1) n 3 2 n 13. ∞ n=2 3 n 4 n−1 14. ∞ n=1 (−1) n 3 n 15. ∞ n=100 10 n 16. Does the series ∞ k=1 ln(k+2) 3k converge or diverge? Explain. 17. Consider the sum q 5 − q 7 + q 9 − q 11 +···+q 41 . (a) Put the sum into closed form. (b) Put the sum into summation notation. (c) Now put −q 5 + q 7 − q 9 + q 11 −···−q 41 into summation notation. 18. For each of the following geometric sums, first write the sum using summation notation and then write the sum in closed form. (a) 2 3 2 + 2 3 4 + 2 3 6 +··· 2 3 18 (b) 1 − 2 + 2 2 − 2 3 + 2 4 −···+2 46 (c) − 1 100 + 1.1 100 − 1.21 100 + 1.331 100 −···− 1.1 100 100 (d) 2 3 2 + 2 2 3 3 + 2 3 3 4 +···+ 2 16 3 17 19. Write the following without using summation notation and answer the following ques- tions. 18.5 Applications of Geometric Sums and Series 579 (a) Is the series geometric? (b) Does the series converge? If so, indicate to what it converges. i) ∞ n=0 3 2 5 2 n ii) ∞ n=0 15 10 2 1 10 3n iii) ∞ n=1 3 2 −2 3 n iv) ∞ n=1 ln n 20. Write the following sums in summation notation. In each case, determine the sum; i.e., sum the first and determine what the second converges to. (a) 500 + 500e .1 + 500e .2 + 500e .3 +···+500e 2 (b) 5 3 − 5 6 + 5 12 − 5 24 +··· 18.5 APPLICATIONS OF GEOMETRIC SUMS AND SERIES ◆ EXAMPLE 18.10 Paulina is self-employed. On the first day of every month she puts $400 into an account she has set up for retirement. The account pays 0.5% per month, i.e., 6% per year compounded monthly. How much money will be in Paulina’s retirement account five years after she sets up the account immediately, before her 61st payment? (Assume that interest is paid on the last day of every month.) SOLUTION Making Estimates. If we ignored interest completely, there would be $400 · 60 = $24,000 in the retirement account. Because there is interest, we expect the answer to be more than $24,000. Each payment is in the bank for a different amount of time. Money in the account grows according to M(t) = M 0 (1.005) 12t or M(t) = M 0 (1.005) m , where t is time in years and m is the number of months the money has been in the bank. If all the money were in the bank for five years she would have $24,000(1.005) 12·5 = $24,000(1.005) 60 = $32, 372.40. Thus, we expect the actual balance to be greater than $24,000 and less than $32,372. Strategy: We will look at each of the 60 payments individually and determine how much each will grow to by the end of the five-year period. In other words, we will look at the future value of each of the 60 payments. Then we’ll sum these future values. We can represent this diagramatically. We’ll push each of the payments to the same point in the future. $400 $400 $400 $400 60th payment 1 month after 60th payment • • • The first payment is in the bank for 60 months, the second for 59 months, the third for 58 months, and so on. The 60th payment is in the bank for 1 month. 580 CHAPTER 18 Geometric Sums, Geometric Series future value of the 1st payment = $400(1.005) 60 future value of the 2nd payment = $400(1.005) 59 future value of the 3rd payment = $400(1.005) 58 future value of the 4th payment = $400(1.005) 57 . . . future value of the 59th payment = $400(1.005) 2 future value of the 60th payment = $400(1.005) 1 The total amount of money in the account after five years is given by 14 $400(1.005) 1 + $400(1.005) 2 + $400(1.005) 3 +···+$400(1.005) 59 + $400(1.005) 60 . Denote this sum by S. “r” = (1.005). Put this sum in closed form. You will get 400(1.005) − 400(1.005) 61 1 − 1.005 or $28, 047.55. Look back at our original expectations. This answer falls in the interval we expected. ◆ ◆ EXAMPLE 18.11 Marietta puts $200 into an account every year for four years in order to finance a long vacation in Greece. The bank pays 5% interest per year compounded annually. Her first payment is January 1, 2000. She estimates that her vacation will cost $900. Will she be able to go on the trip immediately after her fourth payment, on January 1, 2003? Will she be able to go on the trip one year after making her fourth payment? SOLUTION Making Estimates. Certainly if Marietta makes five payments she’ll have over $1000 and the trip won’t be a problem. After making only four payments she will have put a total of $800 into the vacation account. If that total were in the bank for 3 years it would grow to $800(1.05) 3 = $926.10, but the total is not in the bank for three years. Strategy: We’ll find the amount of money in the account on January 1st, 2003. We can represent this diagramatically, pushing all the payments to this date. $200 $200 $200 $200 January 1, 2003 On January 1, 2003 the first payment has grown to $200(1.05) 3 , the second payment has grown to $200(1.05) 2 , the third payment has grown to $200(1.05) 1 , the fourth payment remains at $200. 14 The order in which the addition is done does not matter. Had you chosen to write the sum as $400(1.005) 60 + $400(1.005) 59 + $400(1.005) 58 +···+$400(1.005) 2 + $400(1.005) 1 you would have r = (1.005) −1 . . rate; The partial sums are unbounded. Figure 18.1 If all of the terms of the series are negative, an analogous argument can be made. If some of the terms of an infinite series are positive and others. +···+(2n−1)=n 2 . 12. Give an example of each of the following. (a) An infinite series that converges and whose partial sums are always increasing (b) An infinite series that converges and whose partial sums. actual balance to be greater than $24,000 and less than $32,372. Strategy: We will look at each of the 60 payments individually and determine how much each will grow to by the end of the five-year