Exercise 3.3

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MA111: Prepared by Dr.Archara Pacheenburawana 53 43. 46. 49. 52. ∞∑ n=1 ∞∑ n=1 ∞∑ n=1 10 n (n+1)4 2n+1 44. n n 3 1+3n 47. n n (2n)! 50. ∞∑ n! (−10) n 45. ∞∑ ( ) n 2 n +1 2n 2 +1 48. ∞∑ 4 n +n n! 51. n=1 n=1 n=1 ∞∑ cos(nπ/3) n! ∞∑ 1 2+sin 2 n ∞∑ ( n 3n+2 1 1·2 + 1 2·3 + 1 3·4 + 1 4·5 +··· 53. 1 3 + 2 3 2 + 3 3 3 + 4 3 4 +··· 54. 1+ 1 2 √ 2 + 1 3 √ 3 + 1 4 √ 4 +··· 55. ∞ ∑ 56. 2 1·3·4 + 3 2·4·5 + 4 3·5·6 + 5 4·6·7 +··· n=1 Answer to Exercise 7.3 n=1 n=1 n=1 2·4·6·····(2n) n! 1. C 2. D 3. C 4. C 5. C 6. C 7. C 8. D 9. D 10. C 11. C 12. D 13. D 14. C 15. C 16. C 17. C 18. D 19. C 20. C 21. D 22. C 23. C 24. C 25. D 26. C 27. D 28. C 29. C 30. D 31. D 32. C 33. C 34. D 35. C 36. D 37. C 38. C 39. C 40. C 41. C 42. C 43. C 44. D 45. C 46. D 47. C 48. C 49. C 50. C 51. C 52. C 53. C 54. C 55. D 56. C Exercise 7.4 1. Find the Maclaurin series for the function. (a) e −x (b) e ax (c) cosπx (d) sinπx (e) ln(1+x) 1 (f) 1+x (g) coshx (h) sinhx (i) xsinx (j) xe x 2. Find the Taylor series of the function f(x) centered at the given value of a. (a) e x ; a = 1 (b) e −x ; a = ln2 (c) 1 x ; a = −1 1 (d) x+2 ; a = 3 (e) sinπx; a = 1 (f) cosx; a = π 2 2 (g) lnx; a = 1 (h) lnx; a = e ) n
MA111: Prepared by Dr.Archara Pacheenburawana 54 1. (a) (i) 2. (a) (g) ∞∑ (−1) n x n (c) n! n=0 ∞∑ n=0 ∞∑ n=0 (−1) n (2n+1)! x2n+2 e n! (x−1)n (c) Answer to Selected Exercise 7.4 ∞∑ (−1) n π 2k x 2n (e) (2n)! n=0 ∞∑ (−1) n−1 (x−1) n n n=0 ∞∑ (−1)(x+1) n (e) n=0 ∞∑ (−1) n+1 x n (g) n n=0 ∞∑ (−1) n π 2n n=0 (2n)! ∞∑ n=0 ( x− 1 ) 2n 2 1 (2n)! x2n
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Exercise 3.3

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