Since 0 1 the series converges by the root test 53 1

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Unformatted text preview: n n n 0 ⇒ lim ln n 1 ln n 1 ln n n an ≤ , n 1 1 1 n n 1 lim an 1 an lim n→ lim n→ lim n→ 19 7 n! 1 n n nn n1 1 n 1 n 1 1n 1 <1 e Hence, the series converges. 19 7 n 19 7 n 19 7 n 1 ! 19 nn 1 7 1 1 an . n 1 an ≤ n n k an ≤ an ⇒ an ≤ n k an ≤ 105. Using the Ratio Test, n→ r > R > 1. Then there must exist some M > 0 such that n an > R for infinitely many n > M. Thus, for infinitely many n > M, n we have an > R > 1 which implies that lim an 0 n→ which in turn implies that and choose R such that 0 ≤ r < R < 1. There must exist some N > 0 such that n an < R for all n > N. Thus, for n > N an < Rn and since the geometric series n 471 Second, let n lim The Ratio and Root Tests 1 an . n 1 472 Chapter 9 Infinite Series Section 9.7 12 2x 1. y Taylor Polynomials and Approximations 3. y 1 e 12 x Parabola 1 Linear Matches (d) 1 Matches (a) 4 5. f x fx f1 32 2x P1 x 12 4x x f1 f1 f1x 4 2x P1 x 4 2x 7. f x 2 sec x sec x tan x fx f f 2 4 1 P1 x 1 f f 4 x 4 4 6 P1 x 10 P1 2 4 2 2x 4 5 (1, 4) f f −2 ( π , 2) 4 6 −2 − P1 4 2 −1 P1 is called the first degree Taylor polynomial for f at c. P1 is called the first degree Taylor polynomial for f at c. 4 9. f x fx 12 4x x f1 4 10 P2 32 2x f1 (1, 4) 2 f fx 3x P2 52 f1 f1 f1x 4 2x x f1 x 2 1 3 x 2 1 3 1 −2 6 −2 1 2 2 0.8 0.9 1.0 1.1 1.2 2 fx Error 4.4721 4.2164 4.0 3.8139 3.6515 2.8284 P2 x 11. 0 7.5 4.46 4.215 4.0 3.815 3.66 3.5 fx cos x (b) 1 P6 x 14 24 x 1 P4 x 12 2x 12 2x 14 24 x (a) P4 3 P2 −2 x cos x P2 x 1 P2 0 f sin x P4 cos x 4 x 1 f P6 f P2 x f0 16 720 x 2 −3 sin x fx 1 P2 x fx 12 2x 4 f f 4 f 6 x cos x f 6 0 1 5 x 0 x 1 (c) In general, f P4 P4 sin x n 4 x x x 1 0 P6 5 x x 6 1 P x P6 6 0 0 Pn n 0 for all n. S ection 9.7 fx 13. x e fx x e fx f f0 x P3 x x f f0 e2x f0 1 fx 2e2x f0 2 fx 4e2x f0 4 f x 8e2x f 0 8 x 162x 0 16 1 f0 e fx 15. 1 f0 x e 1 0 1 f02 x 2! f 0x Taylor Polynomials and Approximations f f 03 x 3! 4 4 f fx 17. fx x f f f 5 x x P5 x cos x 0 0 cos x f 0 1 0 02 x 2! 1x fx 4 f 1 x x f 1 x 6 1 2 3 x 1 4 f f 5 fx 25. 1 x x x2 f f x 4 x P4 x x f 0 3 x x x 0 4 4 xe 4e f 4 0 x 22 x 2! 33 x 3! x2 13 x 2 44 x 4! 14 x 6 6 x4 24 x5 f f x 24 x 4! 1 f0 0 sec3 x f0 1 1 sec x tan2 x 12 x 2! 0x 12 x 2 1 24 24 4 x 4! fx 2 x f1 1 fx 27. 1 2x f1 1 2 fx 6 x f 24 2 x 2! 1 sec x tan x 1 1 1 1 6 0 1 4 f0 P2 x 2 sec x fx 1 f1 x 2 3ex 23. f x 0 f1 2 x3 1 f0 xex x4 f1 1 x2 1 1 2ex x 1 63 x 3! x3 1 x fx fx 4 22 x 2! 1 P4 x f0 xex 15 x 5! 1 f0 24 x 04 x 4! f0 2 fx 0 fx 1 x f0 ex P4 x f0 1 fx f 13 x 3! 1 x 24 x 3 xex 15 x 120 13 x 6 43 x 3 f 1 5 2x 2 16 4 x 4! fx 0 f sin x 2x 83 x 3! fx 0 4 x 21. f 42 x 2! xex 1 f0 2x fx 19. 0 f0 sin x x 4 f0 cos x fx P4 x x3 6 sin x 1 1 1 x2 2 1 6 x 3! 2 f 1 P4 x 1 2 x 1 3 x x 3 4 x 4 1 4 1 1 4x x 3 8x 2 x f 15 16x3 x 1 x 2 1 x 16 f 1 x 8 1 1 1 4 f1 3 3 8 1 4 15 16 1 1 5 x 128 2 1 4 473 474 Chapter 9 29. fx ln x f1 0 fx 1 x f1 1 1 x2 fx 4 f f1 2 x3 x f Infinite Series f 6 x4 x P4 x 0 f x 31. 1 4 1 sec2 x fx 2 sec2 x tan x x 4 sec2 x tan2 x 4 tan x fx 6 1 x 12 2 1 x1 4 3 fx 2 1 1 1 x 3 1 f (a) n 3, c P3 x 2 sec4 x f 4 x 8 sec2 x tan3 x f 5 x 16 sec2 x tan4 x (b) n 0 0 3, c 16 sec6 x Q3 x 23 x 3! 4 4 x 2! 13 x 3 x 4 16 sec4 x tan x 88 sec4 x tan2 x 02 x 2! x 1 2x 2 4 3 16 x 3! 4 6 2 1 − P5 2 P3 2x 2x 4 4 8 x 3 2 f Q3 −6 33. fx sin x P1 x x P3 x x 13 6x P5 x x 13 6x (a) 15 120 x (b) x 0.00 0.25 0.50 0.75 1.00 sin x 0.0000 0.2474 0.4794 0.6816 0.8415 P1 x 0.0000 0.2500 0.5000 0.7500 1.0000 P3 x 0.0000 0.2474 0.4792 0.6797 0.8333 P5 x 0.0000 0.2474 0.4794 0.6817 3 0.8417 P1 P3 f −2 2 P5 −3 (c) As the distance increases, the accuracy decreases. 35. f x arcsin x (a) P3 x (b) x x3 6 (c) y π 2 x 0.75 0.50 0.25 0 0.25 0.50 fx 0.848 0.524 0.253 0 0.253 0.524 0.848 P3 x 0.820 0.521 0.253 0 0.253 0.521 0.820 f 0.75 x −1 1 P3 − π 2 3 4 S ection 9.7 37. f x cos x P8 y P4 Taylor Polynomials and Approximations ln x2 39. f x) y 1 6 P6 P2 3 4 2 f (x) = cos x 2 f (x) = ln (x 2 + 1) 1 x −6 6 x −4 −3 −2 8 2 −4 −6 −3 3 4 P4 P8 P6 P2 41. f x e x 1 1 2 fx ln x x3 6 0.6042 f 43. x2 2 x f 1.2 x 1 2 1 x 1 2 1 3 x 1 3 1 4 x 4 0.1823 45. f x 5 cos x; f sin x ⇒ Max on 0, 0.3 is 1. x 1 0.3 5 2.025 10 5 R4 x ≤ 5! Note: you could use R5 x : f 6 x R5 x ≤ 1 0.3 6! 6 1.0125 Exact error: 0.000001 47. f x arcsin x; f R3 x ≤ 49. g x 4 6 10 1.0 10 6 x 6x 2 9 ⇒ Max on 0, 0.4 is f 1 x2 7 2 x 7.3340 0.4 4! cos x, max on 0, 0.3 is 1. 4 0.00782 7.82 1 Rn x ≤ 1 n 0.4 7.3340. 1! 0.3 n By trial and error, n n f 1 < 0.001 10 4. Note: You could use R4. fx 51. ≤ 1 for all x. x 4 10 3. The exact error is 8.5 sin x gn 1 ex 1 ex Max on 0, 0.6 is e0.6 Rn ≤ 3. x 1.8221 0.6 n 1! n 1.8221. 1 By trial and error, n 53. f x f n ln x 1 Rn ≤ n n! 1 x1 x n ⇒ Max on 0, 0.5 is n!. 1 1! 0.5 By trial and error, n ln 1.5 0.4055. n 1 e n 1 9. (See Example 9.) Using 9 terms, Rn ≤ x n x n n 5. f (1.3 e 1 0.5 < 0.0001 n1 x, fx f n n! n 55. f x 1 < 0.001 1e x ≤ 1 1! 1.3 By trial and error, n n 1 < 0.0001 16. n 1 on 0, 1.3 475 476 57. Chapter 9 Infinite Series ex fx 1 x x2 2 x3 ,x<0 6 59. f x ez 4 x < 0.001 4! R3 x n f 4 x x < 0.3936 x< 1 R5 x ez x 4 < 0.024 xe z cos x 6 x2 2! 1 x4 , fifth degree polynomial 4! ...
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This note was uploaded on 01/11/2013 for the course MATH 111 taught by Professor Man during the Spring '13 term at University of Washington-Tacoma Campus.

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