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 f0 ( x ) = a0 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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