09 - CHAPTER Infinite Series Section 9.1 Section 9.2...

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CHAPTER 9 Infinite Series Section 9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Section 9.2 Series and Convergence . . . . . . . . . . . . . . . . . . . 247 Section 9.3 The Integral Test and p -Series . . . . . . . . . . . . . . . . 260 Section 9.4 Comparisons of Series . . . . . . . . . . . . . . . . . . . . 270 Section 9.5 Alternating Series . . . . . . . . . . . . . . . . . . . . . . 277 Section 9.6 The Ratio and Root Tests . . . . . . . . . . . . . . . . . . 286 Section 9.7 Taylor Polynomials and Approximations . . . . . . . . . . 298 Section 9.8 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 308 Section 9.9 Representation of Functions by Power Series . . . . . . . 321 Section 9.10 Taylor and Maclaurin Series . . . . . . . . . . . . . . . . 329 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
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CHAPTER 9 Infinite Series Section 9.1 Sequences 233 1. a 5 5 2 5 5 32 a 4 5 2 4 5 16 a 3 5 2 3 5 8 a 2 5 2 2 5 4 a 1 5 2 1 5 2 a n 5 2 n 2. a 5 5 3 5 5! 5 243 120 5 81 40 a 4 5 3 4 4! 5 81 24 5 27 8 a 3 5 3 3 3! 5 27 6 5 9 2 a 2 5 3 2 2! 5 9 2 a 1 5 3 1! 5 3 a n 5 3 n n ! 3. a 5 5 1 2 1 2 2 5 52 1 32 a 4 5 1 2 1 2 2 4 5 1 16 a 3 5 1 2 1 2 2 3 1 8 a 2 5 1 2 1 2 2 2 5 1 4 a 1 5 1 2 1 2 2 1 1 2 a n 5 1 2 1 2 2 n 4. a 5 32 243 a 4 5 16 81 a 3 8 27 a 2 5 4 9 a 1 2 3 a n 5 1 2 2 3 2 n 5. a 5 5 sin 5 p 2 5 1 a 4 5 sin 2 5 0 a 3 5 sin 3 2 1 a 2 5 sin 5 0 a 1 5 sin 2 5 1 a n 5 sin n 2 6. a 5 5 10 8 5 5 4 a 4 5 8 7 a 3 5 6 6 5 1 a 2 5 4 5 a 1 5 2 4 5 1 2 a n 5 2 n n 1 3 7. a 5 5 s 2 1 d 15 5 2 1 25 a 4 5 s 2 1 d 10 4 2 5 1 16 a 3 5 s 2 1 d 6 3 2 5 1 9 a 2 5 s 2 1 d 3 2 2 1 4 a 1 5 s 2 1 d 1 1 2 1 a n 5 s 2 1 d n s n 1 1 d y 2 n 2 8. a 5 5 2 5 a 4 2 4 1 2 a 3 5 2 3 a 2 2 2 1 a 1 5 2 1 5 2 a n 5 s 2 1 d n 1 1 1 2 n 2 9. a 5 5 5 2 1 5 1 1 25 5 121 25 a 4 5 5 2 1 4 1 1 16 5 77 16 a 3 5 5 2 1 3 1 1 9 5 43 9 a 2 5 5 2 1 2 1 1 4 5 19 4 a 1 5 5 2 1 1 1 5 5 a n 5 5 2 1 n 1 1 n 2
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234 Chapter 9 Infinite Series 10. a 5 5 10 1 2 5 1 6 25 5 266 25 a 4 5 10 1 1 2 1 3 8 5 87 8 a 3 5 10 1 2 3 1 2 3 5 34 3 a 2 5 10 1 1 1 3 2 5 25 2 a 1 5 10 1 2 1 6 5 18 a n 5 10 1 2 n 1 6 n 2 11. 5 2 s 10 2 1 d 5 18 a 5 5 2 s a 4 2 1 d 5 2 s 6 2 1 d 5 10 a 4 5 2 s a 3 2 1 d 5 2 s 4 2 1 d 5 6 a 3 5 2 s a 2 2 1 d 5 2 s 3 2 1 d 5 4 a 2 5 2 s a 1 2 1 d a 1 5 3, a k 1 1 5 2 s a k 2 1 d 12. a 5 5 1 4 1 1 2 2 a 4 5 30 a 4 5 1 3 1 1 2 2 a 3 5 12 a 3 5 1 2 1 1 2 2 a 2 5 6 a 2 5 1 1 1 1 2 2 a 1 5 4 a 1 5 4, a k 1 1 5 1 k 1 1 2 2 a k 13. a 5 5 1 2 a 4 5 1 2 s 4 d 5 2 a 4 5 1 2 a 3 5 1 2 s 8 d 5 4 a 3 5 1 2 a 2 5 1 2 s 16 d 5 8 a 2 5 1 2 a 1 5 1 2 s 32 d 5 16 a 1 5 32, a k 1 1 5 1 2 a k 14. a 5 5 1 3 a 4 2 5 1 3 s 768 d 2 5 196,608 a 4 5 1 3 a 3 2 5 1 3 s 48 2 d 5 768 a 3 5 1 3 a 2 2 5 1 3 s 12 2 d 5 48 a 2 5 1 3 a 1 2 5 1 3 s 6 2 d 5 12 a 1 5 6, a k 1 1 5 1 3 a k 2 15. decreases to 0; matches (f). a n 5 8 n 1 1 , a 1 5 4, a 2 5 8 3 , 16. increases towards 8; matches (a). a n 5 8 n n 1 1 , a 1 5 4, a 2 5 16 3 , 17. decreases to 0; matches (e). a n 5 4 s 0.5 d n 2 1 , a 1 5 4, a 2 5 2, 18. eventually approaches 0; matches (b). a n 5 4 n n ! , a 1 5 4, a 2 5 8, 19. etc.; matches (d). a 3 52 1, a n s 2 1 d n , a 1 1, a 2 5 1, 20. etc.; matches (c). a 3 1 3 , a n 5 s 2 1 d n n , a 1 1, a 2 5 1 2 , 21. a n 5 2 3 n , n 5 1, . . . , 10 1 1 12 8 22. n 5 1, . . . , 10 a n 5 2 2 4 n , 11 2 3 4 23.
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This note was uploaded on 11/13/2010 for the course MATH MAT 231 taught by Professor Thurber during the Spring '08 term at Thomas Edison State.

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09 - CHAPTER Infinite Series Section 9.1 Section 9.2...

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