q2b_sol - MATH 2004. Quiz # 2. A 2-periodic function given...

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MATH 2004. Quiz # 2. A 2 π -periodic function given on the interval ( - π, π ) by the formulas f ( x ) = ± 0 , - π x 0 , x, 0 < x < π Its Fourier series is the following: f ( x ) = π 4 - 2 π X n =1 n - odd 1 n 2 cos( nx ) + X n =1 ( - 1) n +1 n sin( nx ) . ( * ) Is it possible to get the Fourier series for the derivative f 0 ( x ) = a 0 2 + X n =1 ( a n cos( nx ) + b n sin( nx )) doing term-by-term differentiation of the Fourier series ( * )? If yes, find the Fourier coefficient a 3 for the derivative. If not, explain why. A. No, term-by-term differentiation is not valid for the Fourier series of f ( x ) (explain why). B. Yes. For the derivative a 3 = - 2 9 π C. Yes. For the derivative a 3 = 1 3 D. Yes. For the derivative a 3 = 1 E. Yes. For the derivative a 3 = 2 3 π Answer: No, term-by-term differentiation is not valid for the Fourier series of f . Explanation: The function f is
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