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Question 1. (10 marks) (a) Determine the Fourier series of the function f(x) = 4 - x^2 0 < x < 2 extended to an odd function on -2 < x < 2 and then periodically to R. (3 marks) (b) Plot the graphs of f and its odd extension to -2 < x < 2. Furthermore produce three plots of the Fourier series, containing the first 10, 20, and 50 terms, respectively. The plots should have domain -4 ? x ? 4 and range [-5, 5]. If you do not have a computer, handmade sketches are OK. (2 marks) (c) What does this Fourier series converge to? Explain your answer by quoting an appropriate result from the slides. (3 marks) (d) Use your result to compute ?_{k=0}^{?} (-1)^k ( 8 / (?(2k + 1)) + 32 / (?^3(2k + 1)^3) ). (2 marks)

          Question 1. (10 marks)
(a) Determine the Fourier series of the function
f(x) = 4 - x^2 0 < x < 2
extended to an odd function on -2 < x < 2 and then periodically to R. (3 marks)
(b) Plot the graphs of f and its odd extension to -2 < x < 2. Furthermore produce three plots of the Fourier series, containing the first 10, 20, and 50 terms, respectively. The plots should have domain -4 ? x ? 4 and range [-5, 5]. If you do not have a computer, handmade sketches are OK. (2 marks)
(c) What does this Fourier series converge to? Explain your answer by quoting an appropriate result from the slides. (3 marks)
(d) Use your result to compute
?_{k=0}^{?} (-1)^k ( 8 / (?(2k + 1)) + 32 / (?^3(2k + 1)^3) ). (2 marks)

Question 1. (10 marks)
(a) Determine the Fourier series of the function
f(x) = 4 - x^2 0 < x < 2
extended to an odd function on -2 < x < 2 and then periodically to R. (3 marks)
(b) Plot the graphs of f and its odd extension to -2 < x < 2. Furthermore produce three plots of the Fourier series, containing the first 10, 20, and 50 terms, respectively. The plots should have domain -4 ? x ? 4 and range [-5, 5]. If you do not have a computer, handmade sketches are OK. (2 marks)
(c) What does this Fourier series converge to? Explain your answer by quoting an appropriate result from the slides. (3 marks)
(d) Use your result to compute
?k=0^? (-1)^k ( 8 / (?(2k + 1)) + 32 / (?^3(2k + 1)^3) ). (2 marks)

Added by Daryl H.

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