Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
      File size:
      493.6 KB
  2. jcsd
  3. Sep 12, 2011 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook