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Solve the Sine and Cosine Fourier Series and Determine the Sum of Each

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Write the cosine and sine Fourier Series. Determine the sum of each.
{ 4x 0≤x≤2
f(x) ={ -3 2<x<4
{ 1 4&#8804;x&#8804;7

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Sine and Cosine Fourier Series and sums are found. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Hi again;
(1) This part I'll repeat from last time, so that you have it with you:

Two important functions that we'll have to integrate when we try to find the fourier coefficients are:
xsin(kx) and xcos(kx), for some constant k.

We can integrate these by parts, or note that:
xsin(kx) = d/dx ( (-1/k)xcos(kx) + (1/k^2)sin(kx) )
xcos(kx) = d/dx ( (1/k)xsin(kx) + (1/k^2)cos(kx) )

Now, for a function defined on the interval [0,2L], recall that the Fourier ...

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