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to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 35. f(3x) = In (1 - 3x) 37, h(x) = x ln (1-x) 36. g(x) = x³ ln (1-x) 38 BL

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 35. f(3x) = In (1 - 3x) 37, h(x) = x ln (1-x) 36. g(x) = x³ ln (1-x) 38 BL

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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35-40. Combining power series Use the power series representation f(x) = ln (1-x) = - = for -1 ≤ x < 1, to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 35. f(3x) = In (1 - 3x) 37. h(x) = x ln (1-x) 39. p(x) = 2x6 In (1 − x) 36. g(x) = x³ ln (1-x) 38. f(x³) = ln (1 - x³) 40. f(-4x) = ln (1 + 4x) 41-46. Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrat- the power series for f(perhaps more than once). Give the interval of convergence for the resulting series. ing 41. g(x) 42. g(x) 43. g(x) = = (1 - x)² 44. g(x) = 1 (1 - x)³ 1 (1-x) 4 1 using f(x) = 1 - x X using f(x) X (1 + x²)² 45. g(x) = In (1 - 3x) using f(x) 52 Functions 1 using f(x) = -x 1- X 1 1-x 1 - x using f(x) 46. g(x) = In (1 + x²) using f(x) 1 1 + x² = 1 - 3x X 1 + x² 55. Radius of convers 56-59. Summation ne summation (sigma) no 56. 1 + 58. x x + 2 4 1 42 X3 4 power series Find power series representations power series. + + x³ 9 60. Scaling powe interval of cc gence of the 61. Shifting po an interval convergen is a real n 62-67. Series following ser 62. Σ (x2 k=0 X Σ' 64. Σ 66. Σ Σ 68. A la
Transcribed Image Text:35-40. Combining power series Use the power series representation f(x) = ln (1-x) = - = for -1 ≤ x < 1, to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series. 35. f(3x) = In (1 - 3x) 37. h(x) = x ln (1-x) 39. p(x) = 2x6 In (1 − x) 36. g(x) = x³ ln (1-x) 38. f(x³) = ln (1 - x³) 40. f(-4x) = ln (1 + 4x) 41-46. Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrat- the power series for f(perhaps more than once). Give the interval of convergence for the resulting series. ing 41. g(x) 42. g(x) 43. g(x) = = (1 - x)² 44. g(x) = 1 (1 - x)³ 1 (1-x) 4 1 using f(x) = 1 - x X using f(x) X (1 + x²)² 45. g(x) = In (1 - 3x) using f(x) 52 Functions 1 using f(x) = -x 1- X 1 1-x 1 - x using f(x) 46. g(x) = In (1 + x²) using f(x) 1 1 + x² = 1 - 3x X 1 + x² 55. Radius of convers 56-59. Summation ne summation (sigma) no 56. 1 + 58. x x + 2 4 1 42 X3 4 power series Find power series representations power series. + + x³ 9 60. Scaling powe interval of cc gence of the 61. Shifting po an interval convergen is a real n 62-67. Series following ser 62. Σ (x2 k=0 X Σ' 64. Σ 66. Σ Σ 68. A la
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