A.11.3: Section 11.3 Answers

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1. \(C(x)=\frac{L^{2}}{3}+\frac{4L^{2}}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos\frac{n\pi x}{L}\)

2. \(C(x)=\frac{1}{2}+\frac{4}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\cos (2n-1)\pi x\)

3. \(C(x)=-\frac{2L^{2}}{3}+\frac{4L^{2}}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\cos\frac{n\pi x}{L}\)

4. \(C(x)=\frac{1-\cos k\pi}{k\pi}-\frac{2k}{\pi}\sum_{n=1}^{\infty}\frac{[1-(-1)^{n}\cos k\pi}{n^{2}-k^{2}}\cos nx\)

5. \(C(x)=\frac{1}{2}-\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}\cos\frac{(2n-1)\pi x}{L}\)

6. \(C(x)=-\frac{2L^{2}}{3}+\frac{4L^{2}}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos\frac{n\pi x}{L}\)

7. \(C(x)=\frac{1}{3}+\frac{4}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\cos n\pi x\)

8. \(C(x)=\frac{e^{\pi}-1}{\pi }+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{[(-1)^{n}e^{\pi}-1]}{(n^{2}+1)}\cos nx\)

9. \(C(x)=\frac{L^{2}}{6}-\frac{L^{2}}{\pi ^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\cos\frac{2n\pi x}{L}\)

10. \(C(x)=-\frac{2L^{2}}{3}+\frac{4L^{2}}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{n^{2}}\cos\frac{n\pi x}{L}\)

11. \(S(x)=\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{1}{(2n-1)}\sin\frac{(2n-1)\pi x}{L}\)

12. \(S(x)=\frac{2}{\pi} \sum_{n=1}^{\infty}\frac{1}{n}\sin n\pi x\)

13. \(S(x)=\frac{2}{\pi} \sum_{n=1}^{\infty}[1-(-1)^{n}\cos k\pi]\frac{n}{n^{2}-k^{2}}\sin nx\)

14. \(S(x)=\frac{2}{\pi} \sum_{n=1}^{\infty}\frac{1}{n}\left[1-\cos\frac{n\pi}{2}\right]\sin\frac{n\pi x}{L}\)

15. \(S(x)=\frac{4L}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{(2n-1)^{2}}\sin\frac{(2n-1)\pi x}{L}\)

16. \(S(x)=\frac{\pi}{2}\sin x-\frac{16}{\pi} \sum_{n=1}^{\infty}\frac{n}{(4n^{2}-1)^{2}}\sin 2nx\)

17. \(S(x)=-\frac{2}{\pi} \sum_{n=1}^{\infty}\frac{n[(-1)^{n}e^{\pi}-1]}{(n^{2}+1)}\sin nx\)

18. \(C_{M}(x)=-\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}\cos\frac{(2n-1)\pi x}{2L}\)

19. \(C_{M}(x)=-\frac{4L^{2}}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n-1}\left[1-\frac{8}{(2n-1)^{2}\pi ^{2}}\right]\cos\frac{(2n-1)\pi x}{2L}\)

20. \(C_{M}(x)=-\frac{4}{\pi} \sum_{n=1}^{\infty}\left[ (-1)^{n}+\frac{2}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2}\)

21. \(C_{M}(x)=-\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{1}{2n-1}\cos\frac{(2n+1)\pi}{4}\cos\frac{(2n-1)\pi x}{2L}\)

22. \(C_{M}(x)=\frac{4}{\pi} \sum_{n=1}^{\infty}(-1)^{n}\frac{2n-1}{(2n-3)(2n+1)}\cos\frac{(2n-1)x}{2}\)

23. \(C_{M}(x)=-\frac{8}{\pi} \sum_{n=1}^{\infty}\frac{1}{(2n-3)(2n+1)}\cos\frac{(2n-1)x}{2}\)

24. \(C_{M}(x)=-\frac{8L^{2}}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\left[1+\frac{4(-1)^{n}}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

25. \(S_{M}(x)=\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{1}{(2n-1)}\sin\frac{(2n-1)\pi x}{2L}\)

26. \(S_{M}(x)=-\frac{16L^{2}}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\left[ (-1)^{n}+\frac{2}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi x}{2L}\)

27. \(S_{M}(x)=\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{1}{2n-1}\left[ 1-\cos\frac{(2n-1)\pi }{4}\right]\sin\frac{(2n-1)\pi x}{2L}\)

28. \(S_{M}(x)=\frac{4}{\pi} \sum_{n=1}^{\infty}\frac{2n-1}{(2n-3)(2n+1)}\sin\frac{(2n-1)x}{2}\)

29. \(S_{M}(x)=\frac{8}{\pi} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-3)(2n+1)}\sin\frac{(2n-1)x}{2}\)

30. \(S_{M}(x)=\frac{8L^{2}}{\pi ^{2}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{2}}\left[(-1)^{n}+\frac{4}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi x}{2L}\)

31. \(C(x)=-\frac{7L^{4}}{5}-\frac{144L^{4}}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{4}}\cos\frac{n\pi x}{L}\)

32. \(C(x)=-\frac{2L^{4}}{5}-\frac{48L^{4}}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{4}}\cos\frac{n\pi x}{L}\)

33. \(C(x)=\frac{3L^{4}}{5}-\frac{48L^{4}}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{2+(-1)^{n}}{n^{4}}\cos\frac{n\pi x}{L}\)

34. \(C(x)=\frac{L^{4}}{30}-\frac{3L^{4}}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{n^{4}}\cos\frac{2n\pi x}{L}\)

36. \(S(x)=\frac{8L^{2}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\sin\frac{(2n-1)\pi x}{L}\)

37. \(S(x)=-\frac{4L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(1+(-1)^{n}2)}{n^{3}}\sin\frac{n\pi x}{L}\)

38. \(S(x)=-\frac{12L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{3}}\sin\frac{n\pi x}{L}\)

39. \(S(x)=\frac{96L^{4}}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{5}}\sin\frac{(2n-1)\pi x}{L}\)

40. \(S(x)=-\frac{720L^{5}}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{5}}\sin\frac{n\pi x}{L}\)

41. \(S(x)=-\frac{240L^{5}}{\pi ^{5}} \sum_{n=1}^{\infty}\frac{1+(-1)^{n}2}{n^{5}}\sin\frac{n\pi x}{L}\)

43. \(C_{M}(x)=-\frac{64L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

44. \(C_{M}(x)=-\frac{32L^{2}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{3}}\cos\frac{(2n-1)\pi x}{2L}\)

45. \(C_{M}(x)=-\frac{96L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}+\frac{2}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

46. \(C_{M}(x)=\frac{96L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}3+\frac{4}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

47. \(C_{M}(x)=\frac{96L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[(-1)^{n}5+\frac{8}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

48. \(C_{M}(x)=-\frac{384L^{4}}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[1+\frac{(-1)^{n}4}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

49. \(C_{M}(x)=-\frac{768L^{4}}{\pi ^{4}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[ 1+\frac{(-1)^{n}2}{(2n-1)\pi}\right]\cos\frac{(2n-1)\pi x}{2L}\)

51. \(S_{M}(x)=\frac{32L^{2}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\sin\frac{(2n-1)\pi x}{2L}\)

52. \(S_{M}(x)=-\frac{96L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[1+(-1)^{n}\frac{4}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi x}{2L}\)

53. \(S_{M}(x)=\frac{96L^{3}}{\pi ^{3}} \sum_{n=1}^{\infty}\frac{1}{(2n-1)^{3}}\left[1+(-1)^{n}\frac{2}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi x}{2L}\)

54. \(S_{M}(x)=\frac{192L^{3}}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)^{4}}\sin\frac{(2n-1)\pi x}{2L}\)

55. \(S_{M}(x)=\frac{1536L^{4}}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[(-1)^{n}+\frac{3}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi x}{2L}\)

56. \(S_{M}(x)=\frac{384L^{4}}{\pi ^{4}}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^{4}}\left[ (-1)^{n}+\frac{4}{(2n-1)\pi}\right]\sin\frac{(2n-1)\pi x}{2L}\)