SalasSV_11_08_ex_ans - A-80 ANSWERS TO ODD-NUMBERED...

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Unformatted text preview: A-80 ANSWERS TO ODD-NUMBERED EXERCISES (k + 1)k 2 (n + k − 1)! n x + ··· + x + ··· 2! n!(k − 1)! ∞ SECTION 11.8 1. 1 + 2x + 3x2 + · · · + nxn−1 + · · · 5. ln (1 − x2 ) = −x2 − ∞ 3. 1 + kx + 14 16 1 2n+2 x − x − ··· − x − ··· 2 3 n+1 ∞ 7. 1 + x2 + 2 x4 + 3 ∞ 17 6 x 45 + ··· ∞ 9. −72. ( − 1)k 3k +3 x k! 11. k =0 ( − 1)k 4k +2 x (2k + 1)! 13. k =0 3k 3k x k! 27. 15. 2 k =0 ∞ x2k +1 ∞ 17. k =0 (k ! + 1) k x k! ∞ 19. k =1 k ( − 1)k +1 3k +1 x k 21. k =0 23. 1 2 25. − 1 2 k =1 (−1) k2 k −1 x k , −1 ≤ x ≤ 1 29. k =0 (−1) x 2 k +1 (2k + 1)2 41. e x ∞ k =0 3 31. 0. 804 ≤ I ≤ 0. 808 43. 3x2 e x 3 33. 0. 600 ≤ I ≤ 0. 603 35. 0. 294 ≤ I ≤ 0. 304 ∞ 37. 0. 9461 1 39. 0. 4485 1 0 ∞ k =0 45. (a) k =0 1 k +1 x k! (b) 0 xex dx = 1 = 1 k +1 x dx = k! (k ) 1 1 =+ k !(k + 2) 2 ∞ k =1 1 k !(k + 2) 47. Let f (x) be the sum of these series; ak and bk are both f (0)/k !. f (2k −1) (0) = 0 for all k . (2k − 1)! 49. (a) If f is even, then f (2k −1) is odd for k = 1, 2, . . . This implies that f (2k −1) (0) = 0, and so a2k −1 = (b) If f is odd, then f (2k ) is odd for k = 1, 2, . . . , which implies a2k = 0 for all k . 51. f (x) = x − 23 4 8 x + x5 − x7 + · · · = 3! 5! 7! 3 16 ∞ k =0 √ ( − 1)k 2k 2k +1 1 x ; √ sin (x 2) (2k + 1)! 2 55. 0. 2640 ≤ I ≤ 0. 2643; I = 1 − 2/e ∼ 0. 2642411 = 53. 0. 0352 ≤ I ≤ 0. 0359; I = − 3 8 ln 1. 5 ∼ 0. 0354505 = SECTION 11.9 1. 1 + 1 x − 1 x2 + 2 8 9. 8 + 3x + 15. 2. 0799 32 x 16 13 x 16 − 54 x 128 3. 1 + 1 x2 − 1 x4 2 8 ∞ 5. 1 − 1 x + 3 x2 − 2 8 −1/2 k ∞ 53 x 16 + 35 4 x 128 7. 1 − 1 x − 4 R=1 32 x 32 − 73 x 128 − 77 4 x 2048 − 13 x 128 + 3 x4 4096 11. (a) k =0 (−1)k x 2k (b) k =0 (−1)k −1/2 k 1 x2k +1 , 2k + 1 13. 9. 8995 17. 0. 4925 19. 0. 3349 21. 0. 4815 CHAPTER 12 SECTION 12.1 1. z 3. z 5. z = −2 7. y = 1 9. x = 3 A(0,–2,5) A(2,0,0) B (0,0,– 4) x y B (4,1,0) y length AB: 2 √5 midpoint: (1,0,–2) x length AB: 5 √2 midpoint: (2, – 1 , 5 ) 22 11. x2 + ( y − 2)2 + (z + 1)2 = 9 13. (x − 2)2 + ( y − 4)2 + (z + 4)2 = 36 19. center (−2, 4, 1), radius 4 31. (2, −5, 5) 33. (−2, 1, −3) 15. (x − 3)2 + ( y − 2)2 + (z − 2)2 = 13 21. center: (3, −5, 1); radius: 6 23. (2, 3, −5) 25. (−2, 3, 5) 17. (x − 2)2 + ( y − 3)2 + (z + 4)2 = 25 27. (−2, 3, −5) 29. (−2, −3, −5) 35. (x − 3)2 + ( y − 3)2 + (z − 3)2 = 9, (x − 7)2 + ( y − 7)2 + (z − 7)2 = 49 37. not a sphere; the equation is equivalent to (x − 2)2 + (y + 2)2 + (z + 3)2 = −3 ...
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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