29: Section 8.3 Answers

1. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}(2m+1)x^{2m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}(m+1)x^{2m+1}\)

2. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m+1}\frac{x^{2m}}{2m-1}+a_{1}x\)

3. \(y=a_{0}(1-10x^{2}+5x^{4})+a_{1}\left(x-2x^{3}+\frac{1}{5}x^{5}\right)\)

4. \(y=a_{0}\sum_{m=0}^{\infty}(m+1)(2m+1)x^{2m}+\frac{a_{1}}{3}\sum_{m=0}^{\infty}(m+1)(2m+3)x^{2m+1}\)

5. \(y=a_{0}\left(1-14x^{2}+\frac{35}{3}x^{4} \right)+a_{1}\left(x-3x^{3}+\frac{3}{5}x^{5}+\frac{1}{35}x^{7} \right)\)

6. \(y=a_0(1+{1\over 6}x^3+{1\over 180}x^6+\cdots)+a_1(x+{1\over 12}x^4+{1\over 504}x^7+\cdots)\)

7. \(y=a_0(1-{1\over 2}x^2-{1\over 8}x^4+\cdots)+a_1(x+{1\over 6}x^3+{1\over 24}x^5+\cdots)\)

8. \(y=a_0(1-{1\over 6}x^3+{1\over 45}x^6+\cdots)+a_1(x-{1\over 6}x^4+{5\over 252}x^7+\cdots)\)

9. \(y=a_0+a_1\sum_{m=1}^{\infty}{x^m\over m}\)

10. \(y=a_0(1+{1\over 2}x^2+{1\over 6}x^3+\cdots)+a_1(x+{1\over 2}x^2+{1\over 2}x^3+\cdots)\)

11. \(y=a_0(1+{1\over 4}x^2-{7\over 96}x^4+\cdots)+a_1(x-{1\over 6}x^3+{7\over 120}x^5+\cdots)\)

12. \(y=2-x-x^{2}+\frac{1}{3}x^{3}+\frac{5}{12}x^{4}-\frac{1}{6}x^{5}-\frac{17}{72}x^{6}+\frac{13}{126}x^{7}+\ldots \)

13. \(y=1-x+3x^{2}-\frac{5}{2}x^{3}+5x^{4}-\frac{21}{8}x^{5}+3x^{6}-\frac{11}{16}x^{7}+\ldots \)

14. \(y=2-x-2x^{2}+\frac{1}{3}x^{3}+3x^{4}-\frac{5}{6}x^{5}-\frac{49}{5}x^{6}+\frac{45}{14}x^{7}+\ldots \)

15. \(y = 2 − 3x − 2x^{2} + \frac{7}{2}x^{3} − \frac{55}{12}x^{4} + \frac{59}{8}x^{5} − \frac{83}{6}x^{6} + \frac{9547}{ 336}x^{7} + \ldots\)

16. \(y = −1 + 2x − 4x^{3} + 4x^{4} + 4x^{5} − 12x^{6} + 4x^{7} + \ldots \)

17. \(y = 1 + x^{2} − \frac{2}{3}x^{3} + \frac{11}{6}x^{4} − \frac{9}{5} x^{5} + \frac{329}{90}x^{6} − \frac{1301}{315} x^{7} + \dots\)

18. \(y = x − x^{2} − \frac{7}{2} x^{3} + \frac{15}{2} x^{4} + \frac{45}{8} x^{5} − \frac{261}{8}x^{6} + \frac{207}{16} x^{7} + \ldots\)

19. \(y = 4 + 3x − \frac{15}{4} x^{2} + \frac{1}{4} x^{3} + \frac{11}{16} x^{4} − \frac{5}{16} x^{5} + \frac{1}{20} x^{6} + \frac{1}{120} x^{7} + \ldots\)

20. \(y = 7 + 3x − \frac{16}{3} x^{2} + \frac{13}{3} x^{3} − \frac{23}{9} x^{4} + \frac{10}{9} x^{5} − \frac{7}{27} x^{6} − \frac{1}{9} x^{7} + \ldots\)

21. \(y = 2 + 5x − \frac{7}{4} x^{2} − \frac{3}{16} x^{3} + \frac{37}{192} x^{4} − \frac{7}{192} x^{5} − \frac{1}{1920} x^{6} + \frac{19}{11520} x^{7} + \ldots\)

22. \(y=8x-2e^x\)

23. \(3-12x^2+4x^4\)

24. \(y=a_{0}\sum_{m=0}^{\infty}\frac{(x-3)^{2m}}{(2m)!}+a_{1}\sum_{m=0}^{\infty}\frac{(x-3)^{2m+1}}{(2m+1)!}\)

25. \(y=a_{0}\left(1-6(x-2)^{2}+\frac{4}{3}(x-2)^{4}+\frac{8}{135}(x-2)^{6}\right)+a_{1}\left((x-2)-\frac{10}{9}(x-2)^{3}\right)\)

26. \(y=-1+2x+\frac{3}{8}x^{2}-\frac{1}{3}x^{3}-\frac{3}{128}x^{4}-\frac{1}{1024}x^{6}+\ldots \)

27. \(y = −2 + 3(x − 3) + 3(x − 3)^{2} − 2(x − 3)^{3} − \frac{5}{4}(x − 3)^{4} + \frac{3}{5}(x − 3)^{5} + \frac{7}{24}(x − 3)^{6} − \frac{4}{35}(x − 3)^{7} + \ldots \)

28. \(y = −1 + (x − 1) + 3(x − 1)^{2} − \frac{5}{2}(x − 1)^{3} − \frac{27}{4}(x − 1)^{4} + \frac{21}{4}(x − 1)^{5} + \frac{27}{2}(x − 1)^{6} − \frac{81}{8}(x − 1)^{7} + \ldots \)

29. \(y = 4 − 6(x − 3) − 2(x − 3)^{2} + (x − 3)^{3} + \frac{3}{2}(x − 3)^{4} − \frac{5}{4}(x − 3)^{5} − \frac{49}{20}(x − 3)^{6} + \frac{135}{56}(x − 3)^{7} + \dots\)

30. \(y = 3 − 4(x − 4) + 15(x − 4)^{2} − 4(x − 4)^{3} + \frac{15}{4}(x − 4)^{4} − \frac{1}{5}(x − 4)^{5}\)

31. \(y = 3 − 3(x + 1) − 30(x + 1)^{2} + \frac{20}{3}(x + 1)^{3} + 20(x + 1)^{4} − \frac{4}{3}(x + 1)^{5} − \frac{8}{9} (x + 1)^{6}+ \dots\)

32. \(y = 1 − (x − 1) + \frac{4}{3} (x − 1)^{3} − \frac{4}{3} (x − 1)^{4} − \frac{4}{5} (x − 1)^{5} + \frac{136}{45} (x − 1)^{6} − \frac{104}{63} (x − 1)^{7} + \ldots \)

33. \(y = 1 − (x + 1) + 4(x + 1)^{2} − \frac{13}{3} (x + 1)^{3} + \frac{77}{6} (x + 1)^{4} − \frac{278}{15} (x + 1)^{5} + \frac{1942}{45} (x + 1)^{6} − \frac{23332}{315} (x + 1)^{7} + \ldots\)

34. \(y = 2 − (x − 1) − \frac{1}{2} (x − 1)^{2} + \frac{5}{3} (x − 1)^{3} − \frac{19}{12} (x − 1)^{4} + \frac{7}{30} (x − 1)^{5} + \frac{59}{45} (x − 1)^{6} − \frac{1091}{630} (x − 1)^{7} + \ldots\)

35. \(y = −2 + 3(x + 1) − \frac{1}{2}(x + 1)^{2} − \frac{2}{3}(x + 1)^{3} + \frac{5}{8}(x + 1)^{4} − \frac{11}{30} (x + 1)^{5} + \frac{29}{144} (x + 1)^{6} − \frac{101}{840 }(x + 1)^{7} + \ldots\)

36. \(y = 1 − 2(x − 1) − 3(x − 1)^{2} + 8(x − 1)^{3} − 4(x − 1)^{4} − \frac{42}{5}(x − 1)^{5} + 19(x − 1)^{6} − \frac{604}{35} (x − 1)^{7} + \ldots\)

37. \(y = 2 − 7x − 4x^{2} − \frac{17}{6}x^{3} − \frac{3}{4}x^{4} − \frac{9}{40}x^{5} + \ldots\)

38. \(y = 1 − 2(x − 1) + \frac{1}{2} (x − 1)^{2} − \frac{1}{6}(x − 1)^{3} + \frac{5}{36}(x − 1)^{4} − \frac{73}{1080}(x − 1)^{5} + \ldots\)

39. \(y = 2 − (x + 2) −\frac{7}{2}(x + 2)^{2} +\frac{4}{3}(x + 2)^{3} −\frac{1}{24}(x + 2)^{4} +\frac{1}{60}(x + 2)^{5} + \ldots\)

40. \(y = 2 − 2(x + 3) − (x + 3)^{2} + (x + 3)^{3} −\frac{11}{12}(x + 3)^{4} +\frac{67}{60}(x + 3)^{5} +\ldots\)

41. \(y = −1 + 2x + \frac{1}{3}x^{3} −\frac{5}{12}x^{4} + \frac{2}{5}x^{5} + \ldots\)

42. \(y = 2 − 3(x + 1) + \frac{7}{2} (x + 1)^{2} − 5(x + 1)^{3} + \frac{197}{24}(x + 1)^{4} − \frac{287}{20}(x + 1)^{5} + \ldots\)

43. \(y = −2 + 3(x + 2) − \frac{9}{2}(x + 2)^{2} +\frac{11}{6}(x + 2)^{3} +\frac{5}{24}(x + 2)^{4} +\frac{7}{20}(x + 2)^{5} + \ldots\)

44. \(y = 2 − 4(x − 2) − \frac{1}{2}(x − 2)^{2} + \frac{2}{9}(x − 2)^{3} + \frac{49}{432}(x − 2)^{4} + \frac{23}{1080} (x − 2)^{5} + \ldots\)

45. \(y = 1 + 2(x + 4) − \frac{1}{6}(x + 4)^{2} −\frac{10}{27}(x + 4)^{3} +\frac{19}{648}(x + 4)^{4} +\frac{13}{324}(x + 4)^{5} + \ldots\)

46. \(y = −1 + 2(x + 1) −\frac{1}{4}(x + 1)^{2} +\frac{1}{2}(x + 1)^{3} −\frac{65}{96}(x + 1)^{4} + \frac{67}{80}(x + 1)^{5} + \ldots\)

47. \(y = 1 − 2x −\frac{3}{2} x^{2} + \frac{5}{3} x^{3} + \frac{17}{24} x^{4} − \frac{11}{20} x^{5} + \ldots\)

48. \(y = 1 − 2x − \frac{5}{2} x^{2} + \frac{2}{3} x^{3} − \frac{3}{8} x^{4} + \frac{1}{3} x^{5} + \ldots\)

49. \(y = 6 − 2x + 9x^{2} + \frac{2}{3} x^{3} − \frac{23}{4} x^{4} − \frac{3}{10} x^{5} + \ldots\)

50. \(y = 2 − 5x + 2x^{2} − \frac{10}{3} x^{3} + \frac{3}{2} x^{4} − \frac{25}{12} x^{5} + \ldots\)

51. \(y = 3 + 6x − 3x^{2} + x^{3} − 2x^{4} − \frac{17}{20} x^{5} + \ldots\)

52. \(y = 3 − 2x − 3x^{2} + \frac{3}{2} x^{3} + \frac{3}{2} x^{4} − \frac{49}{80} x^{5} + \ldots\)

53. \(y = −2 + 3x + \frac{4}{3} x^{2} − x^{3} − \frac{19}{54} x^{4} + \frac{13}{60} x^{5} + \ldots\)

54. \(y = −2 + 3x + x^{2} − \frac{1}{6} x^{3} −\frac{3}{4}x^{4} + \frac{31}{120}x^{5} + \ldots\)

55. \(y = 2 + 3x − \frac{7}{2} x^{2} − \frac{5}{6} x^{3} + \frac{41}{24} x^{4} + \frac{41}{120} x^{5} + \ldots\)

56. \(y = −3 + 5x − 5x^{2} + \frac{23}{6}x^{3} − \frac{23}{12}x^{4} + \frac{11}{30} x^{5} + \ldots\)

57. \(y = −2 + 3(x − 1) + \frac{3}{2}(x − 1)^{2} −\frac{17}{12}(x − 1)^{3} −\frac{1}{12}(x − 1)^{4} +\frac{1}{8}(x − 1)^{5} + \ldots\)

58. \(y = 2 − 3(x + 2) + \frac{1}{2}(x + 2)^{2} − \frac{1}{3}(x + 2)^{3} + \frac{31}{24}(x + 2)^{4} −\frac{53}{120}(x + 2)^{5} + \ldots\)

59. \(y = 1 − 2x + \frac{3}{2} x^{2} − \frac{11}{6} x^{3} + \frac{15}{8} x^{4} − \frac{71}{60} x^{5} + \ldots\)

60. \(y = 2 − (x + 2) − \frac{7}{2}(x + 2)^{2} − \frac{43}{6}(x + 2)^{3} − \frac{203}{24} (x + 2)^{4} − \frac{167}{30} (x + 2)^{5} + \ldots\)

61. \(y = 2 − x − x^{2} + \frac{7}{6} x^{3} − x^{4} + \frac{89}{120} x^{5} + \ldots\)

62. \(y = 1 + \frac{3}{2} (x − 1)^{2} + \frac{1}{6} (x − 1)^{3} − \frac{1}{8} (x − 1)^{5} + \ldots\)

63. \(y = 1 − 2(x − 3) + \frac{1}{2}(x − 3)^{2} − \frac{1}{6}(x − 3)^{3} + \frac{1}{4}(x − 3)^{4} − \frac{1}{6}(x − 3)^{5} + \ldots\)

64.  a. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}x^{2m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}x^{2m+1}\)
      b. \(y=\frac{a_{0}+a_{1}x}{1+x^{2}}\)

65. \(y_{1}=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m}}{m!}=e^{-x^{2}},\:y_{2}=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m+1}}{m!}=xe^{-x^{2}}\)

66. \(y=a_0(1+{1\over 6}x^3+{1\over 180}x^6+\cdots)+a_1(x+{1\over 12}x^4+{1\over 504}x^7+\cdots)+({1\over 2}x^2+{1\over 40}x^5+{1\over 2240}x^8+\cdots)\)

67. \(y=a_0(1+{1\over 2}x^2-{1\over 24}x^4+\cdots)+a_1x+({1\over 2}x^2+{1\over 6}x^3-{1\over 120}x^5+\cdots)\)

---