5128_Ch08_434-471

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Section 8.1 Sequences 443 56. Fibonacci Sequence The Fibonacci Sequence can be defined recursively by a 1 1, a 2 1, and a n a n2 a n1 for all integers n 3. (a) Write out the first 10 terms of the sequence. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (b) Draw a graph of the sequence using the Sequence Graphing mode on your grapher. Enter u(n) u(n 1) u(n 2) and u(nMin) {1, 1}. Extending the Ideas 57. Writing to Learn If {a n } is a geometric sequence with all positive terms, explain why {log a n } must be arithmetic. 58. Writing to Learn If {a n } is an arithmetic sequence, explain why {10 a n} must be geometric. 59. Proving Limits Use the formal definition of limit to prove that lim 1 n→ n 0. 1. 12, 23, 34, 45, 56, 67; 5051 2. 2, 52, 83, 114, 145, 176; 149/50 3. 2, 94, 6427, 625256, 77763125 2.48832, 11764946656 2.521626; (5150) 50 2.691588 11. (a) 3 (b) 19 (c) a n a n1 3 (d) a n 3n 5 12. (a) –2 (b) 1 (c) a n a n1 2 (d) a n 2n 17 13. (a) 1/2 (b) 92 (c) a n a n1 12 (d) a n (n 1)2 14. (a) 0.1 (b) 3.7 (c) a n a n1 0.1 (d) a n 0.1n 2.9 15. (a) 12 (b) 8(12) 8 0.03125 (c) a n (12)a n1 (d) a n 8(12) n1 2 4n 16. (a) 1.5 (b) (1)(1.5) 8 25.6289 (c) a n (1.5)a n1 (d) a n (1)(1.5) n1 (1.5) n1 17. (a) –3 (b) (3) 9 19,683 (c) a n (3)a n1 (d) a n (3)(3) n1 (3) n 18. (a) –1 (b) (5)(1) 8 5 (c) a n a n1 (d) a n 5(1) n1 57. a n ar n1 implies that log a n log a (n 1) log r. Thus {log a n } is an arithmetic sequence with first term log a and common ratio log r. 58. a n a (n 1)d implies that 10 a n 10 a(n1)d 10 a (10 d ) n1 . Thus {10 a n } is a geometric sequence with first term 10 a and common ratio 10 d . 59. Given e 0 choose M 1e. Then 1 n 0 e if n M.
444 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals 8.2 What you’ll learn about • Indeterminate Form 00 • Indeterminate Forms , • 0, • Indeterminate Forms 1 , 0 0 , 0 . . . and why Limits can be used to describe the behavior of functions and l’Hôpital’s Rule is an important technique for finding limits. L’Hôpital’s Rule Indeterminate Form 00 If functions f x and gx are both zero at x a, then f x lim x→a g x cannot be found by substituting x a. The substitution produces 00, a meaningless expression known as an indeterminate form. Our experience so far has been that limits that lead to indeterminate forms may or may not be hard to find algebraically. It took a lot of analysis in Exercise 75 of Section 2.1 to find lim x→0 sin xx. But we have had remarkable success with the limit f a lim f x f a , x→a x a from which we calculate derivatives and which always produces the equivalent of 00. L’Hôpital’s Rule enables us to draw on our success with derivatives to evaluate limits that otherwise lead to indeterminate forms. THEOREM 4 L’Hôpital’s Rule (First Form) a (x, f(x)) rise = f(x) run = x– a x x Suppose that f a ga 0, that f a and ga exist, and that ga 0. Then f x f a lim . x→a g x g a rise = g(x) (x, g(x)) Figure 8.5 A zoom-in view of the graphs of the differentiable functions f and g at x a. (Theorem 4) Bernard A. Harris Jr. (1956–1969) Bernard Harris, M.D., became an astronaut in 1991. In 1995, as the Payload Commander on the STS-63 mission, he became the first African American to walk in space. During this mission, which included a rendezvous with the Russian Space Station, Mir, Harris traveled over 2.9 million miles. Dr. Harris left NASA in 1996 to become Vice President of Microgravity and Life Sciences for SPACEHAB Incorporated. Proof Graphical Argument If we zoom in on the graphs of f and g at a, f a a, ga a, 0, the graphs (Figure 8.5) appear to be straight lines because differentiable functions are locally linear. Let m 1 and m 2 be the slopes of the lines for f and g, respectively. Then for x near a, f x f x x a m . 1 gx g x m 2 x a As x→a, m 1 and m 2 approach f a and ga, respectively. Therefore, f x m lim lim 1 f a . x→a g x x→a m 2 g a Confirm Analytically Working backward from f a and ga, which are themselves limits, we have lim f x f a f a x→a x a g a lim gx ga x→a x a lim x→a f x f a x a gx ga x a f x f a f x 0 f x lim lim lim . ■ x→a g x ga x→a g x 0 x→a g x
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