MATH 2004. Quiz # 2.A 2π-periodic function given on the interval (-π, π) by the formulasf(x) =0,-π≤x≤0,x,0< x < πIts Fourier series is the following:f(x) =π4-2π∞n=1n-odd1n2cos(nx) +∞n=1(-1)n+1nsin(nx).(*)Is it possible to get the Fourier series for the derivativef(x) =a02+∞n=1(ancos(nx) +bnsin(nx))doing term-by-term differentiation of the Fourier series (*)? If yes, find the Fourier coefficienta3for the derivative. If not, explain why.A. No, term-by-term differentiation is not valid for the Fourier series off(x) (explain why).B. Yes. For the derivativea3=-29πC. Yes. For the derivativea3=13D. Yes. For the derivativea3= 1E. Yes. For the derivativea3=23πAnswer:No, term-by-term differentiation is not valid for the Fourier series off.Explanation:The function
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