Find at least three nonzero terms (including a 0 and at least two cosine terms a

Question

Find at least three nonzero terms (including a 0 and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given function and sketch at least three periods of the function. Which of the following is the Fourier series for the given function? f(x) = 15 pi/4 - 30/pi (cos x + 1/9 cos 3x + ...) - 15 (sin x - 1/2 sin 2x + ...) f(x) = 15 pi/4 - 30/pi (cos x + 1/9 cos 3x + ...) + 15 (sin x - 1/2 sin 2x + ...) f(x) = 15 pi/4 - 30/pi (sin x + 1/9 sin 3x + ...) - 15 (cos x - 1/2 cos 2x + ...) f(x) = 15 pi/4 - 30/pi (cos x + 1/3 sin 3x + ...) - 15 (sin x - 1/2 sin 2x + ...)

Explanation / Answer

a0 = 1/pi integral ( 15x dx)

= 15 pi/2

an = 1/pi integral ( 15x cos(nx) dx)

= 1/pi * 15 [ cos(n pi) - 1] /n^2

bn = 1/pi integral ( 15x sin(nx) dx)

= - 1/pi * 15 pi cos(n pi) /n

hence fourier series = option B

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