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10.3.5 An Interlude Contemporary Calculus 1 10.3.5 AN INTERLUDE The previous three sections introduced the topics of sequences and series, discussed the meaning of convergence of series, and examined geometric series in some detail. The ideas, definitions, and results in those sections are fundamental for understanding and working with the material in the rest of this chapter and for later work in theoretical and applied mathematics. The material in the next several sections is of a different sort –– it is more technical and specialized. In order to work effectively with power series we need to know where (for which values of x) the power series converge. And to determine that convergence we need additional methods. In the next several sections we examine several methods for determining where particular series converge or diverge. These methods are called "convergence tests." • The Integral Test in Section 10.4 says that a series converges if and only if a certain related improper integral is finite. This result lets us change a question about convergence of a series into a question about the convergence of an integral. Sometimes the related integral is easy to evaluate so it is easy to determine the convergence of the series. (Sometimes the related integral is very difficult to evaluate.) The integral test is then used to determine the convergence of P–series, the whole family of series of the form ∑k=1∞ 1kp . • Section 10.5 introduces some methods for determining the convergence of a new series by comparing the new series with some series which we already know converge or diverge. These comparison tests can be very powerful and useful, but their power and usefulness depends on already knowing about the convergence of some particular series to compare against the new series. Typically we will compare new series against two types of series, geometric series and P–series. • In Section 10.6 we derive a result about the convergence of a series ∑k=1∞ ak by examining the ratios of successive terms of the series, ak+1/ak . If this ratio is small enough, then we will be able to conclude that the series converges. If the ratio is large enough, then we will be able to conclude that the series diverges. Unfortunately, sometimes the value of the ratio will not allow us to conclude anything. • Section 10.7 examines series whose terms alternate in sign, such as the "alternating harmonic series," 1 – 1/2 + 1/3 – 1/4 + . . . , and discusses methods to determine whether these alternating series converge.10.3.5 An Interlude Contemporary Calculus 2 Each of these sections is rather short and focuses on one or two tests of convergence. As you study the material in each section by itself, you need to be able to use the method discussed in that section. When you finish all of the sections, you also need to be able to decide which convergence test to use. PROBLEMS These problems illustrate some of the reasoning that is used in sections 10.4 – 10.7, but they do not assume any information from those sections. Integrals and sums 1. Which shaded region in Fig. 1 has the larger area, the sum or the integral? 2. Which shaded region in Fig. 2 has the larger area, the sum or the integral? 3. Represent the area of the shaded region in Fig. 3 as an infinite series. 4. Represent the area of the shaded region in Fig. 4 as an infinite series. 5. Represent the area of the shaded region in Fig. 5 as an infinite series. 6. Which of the following represents the shaded area in Fig. 6? (a) f(0) + f(1) (b) f(1) + f(2) (c) f(2) + f(3) (d) f(3) + f(4) Fig. 11 2 3 4f(x)f(1) + f(2) + f(3)1 2 3 4f(x)!14f(x) dx Fig. 21 2 3 4f(x)f(2) + f(3) + f(4)1 2 3 4f(x)!14f(x) dx Fig. 31 2 3 4 5f(x). . . Fig. 41 2 3 4 5f(x). . . Fig. 51 2 3 4 5f(x). . . 1 2 3 4Fig. 6f(x)10.3.5 An Interlude Contemporary Calculus 3 7. Which of the following represents the shaded area in Fig. 7? (a) f(0) + f(1) (b) f(1) + f(2) (c) f(2) + f(3) (d) f(3) + f(4) 8. Which of the following represents the shaded area in Fig. 8? (a) f(0) + f(1) (b) f(1) + f(2) (c) f(2) + f(3) (d) f(3) + f(4) 9. Which of the following represents the shaded area in Fig. 9? (a) f(0) + f(1) (b) f(1) + f(2) (c) f(2) + f(3) (d) f(3) + f(4) 10. Arrange the following four values in increasing order (Fig. 10): (a) ⌡⌠13 f(x) dx (b) ⌡⌠24 f(x) dx (c) f(1) + f(2) (d) f(2) + f(3) 11. Arrange the following four values in increasing order (Fig. 11): (a) ⌡⌠14 f(x) dx (b) ⌡⌠25 f(x) dx (c) f(1) + f(2) + f(3) (d) f(2) + f(3) + f(4) 1 2 3 4Fig. 7f(x) 1 2 3 4Fig. 8f(x) 1 2 3 4Fig. 9f(x) Fig. 101 3 4 5f(x) Fig. 111 3 4 5f(x)10.3.5 An Interlude Contemporary Calculus 4 Comparisons 12. You want to get a summer job operating a type of heavy equipment, and you know there are certain height requirements in order for the operator to fit safely in the cab of the machine. You don't remember what the requirements are, but three of your friends applied. Tom was rejected as too tall. Sam was rejected as too short. Justin got a job. Should you apply for the job if (a) you are taller than Tom? Why? (b) you are taller than Sam? Why? (c) you are shorter than Sam? Why? (d) you are shorter than Justin? Why? (e) List the comparisons which indicate that you are the wrong height for the job. (f) List the comparisons which do not give you enough information about whether you are an acceptable height for the job. 13. You know Wendy did well on the Calculus test and Paula did poorly, but you haven't received your test back yet. If the instructor tells you the following, what can you conclude? (a) "You did better than Wendy." (b) "You did better than Paula." (c) "You did worse than Wendy." (d) "You did worse than Paula." 14. You have recently taken up mountain climbing and are considering a climb of Mt. Baker. You know that Mt. Index is too easy to be challenging, but that Mt. Liberty Bell is too difficult for you. Should you plan a climb of Mt. Baker if an experienced climber friend tells you that (a) "Baker is easier than Index." (b) "Baker is more difficult than Index." (c) "Baker is easier than Liberty Bell." (d) "Baker is more difficult than Liberty Bell." (e) Which comparisons indicate that Baker is appropriate: challenging but not too difficult?


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BC MATH 153 - AN INTERLUDE

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