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Fourier series convergence test

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data

    A function f(x) is given as follows

    f(x) = 0, , -pi <= x <= pi/2
    f(x) = x -pi/2 , pi/2 < x <= pi

    determine if it's fourier series (given below)

    [itex]F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx)
    - (1/(2n^{2}))( n\pi cos(n\pi) + 2sin(n\pi/2) )sin(nx) ][/itex]

    for n= 1 to infinity


    converges to it for the case of x = pi/2

    2. Relevant equations

    F(x) = 0.5*[f(x+) + f(x-) ]


    3. The attempt at a solution

    see pdf attachment
     

    Attached Files:

    • q37a.pdf
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  2. jcsd
  3. Sep 12, 2011 #2

    vela

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    Your work is kind of hard to follow, but it looks like you incorrectly got rid of the cosine terms. You have

    [tex]F(x)=\frac{\pi}{16} + \frac{1}{\pi}\sum_{n=1}^\infty \frac{\cos n\pi -\cos \frac{n\pi}{2}}{n^2} \cos nx - \frac{1}{\pi}\sum_{n=1}^\infty \frac{n\pi \cos n\pi + 2\sin \frac{n\pi}{2}}{2n^2}\sin nx [/tex]

    You can simplify it a bit by noting that [itex]\cos n\pi = (-1)^n[/itex]. Next, when [itex]x=\pi/2[/itex], the values of cos nx and sin nx depend on whether n is odd or even. Use those facts to get the series for [itex]F(\pi/2)[/itex].
     
    Last edited: Sep 12, 2011
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