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Find the Fourier series for the periodic function

  1. Dec 14, 2016 #1
    < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown >

    Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong.

    Anyway, I have this question...
    Find the Fourier series for the periodic function

    f(x) = x^2 (-pi < x < pi), f(x) = f(x+2pi)

    By considering the particular values x = 0 and x = +/- pi prove that ​

    ∞^Σ (n=1) 1 / n^2 = pi^2 / 6

    and

    ∞^Σ (n=1) (-1)^n / n^2 = -pi^2 / 12

    I missed a couple of lectures due to illness so I'm not entirely sure what to do with this.

    But I have worked through and found the FS itself. Whether it's right I really don't know...

    = pi^2 / 3 + ∞^Σ (n=1) 4/n (-1)^n Cos(nx)


    Again, I'm really sorry that it's all a bit messy but I'm still getting use to everything. I would like to learn how to use LaTeX to make equations easier/clearer also. But, regardless, I would really appreciate some guidance on how to proceed through the question. Many thanks


     
    Last edited by a moderator: Dec 14, 2016
  2. jcsd
  3. Dec 14, 2016 #2

    LCKurtz

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    Your Fourier Series is correct. So you have$$
    x^2 = \frac {\pi^2} 3 + \sum_{n=1}^\infty \frac{4(-1)^n}{n^2}\cos(nx)$$on the interval ##[-\pi,\pi]##. Try plugging in the two values to see if you can arrange it to get your results.
     
    Last edited: Dec 17, 2016
  4. Dec 17, 2016 #3

    vela

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    It's probably just a typo, but the Fourier series should be
    $$\frac{\pi^2}{3}+\sum_{n=1}^\infty \frac{4(-1)^n}{n^2}\cos nx.$$ The ##n## in the denominator is squared.
     
  5. Dec 17, 2016 #4

    LCKurtz

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    Yes, thanks. Fixed but likely irrelevant given that the OP never returned.
     
  6. Jan 9, 2017 #5
    Yeah sorry guys. I thought I'd marked it as solved, and so hadn't been on the forums. I had made a mistake on the denominator but I noticed and fixed it. Thank you both
     
  7. Jan 9, 2017 #6

    Ray Vickson

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    If you are going to use plain text, it is better to write your summations as something like sum_{n=1 ..∞} a_n or sum(a_n: n=1..∞), and you can, of course, replace the word "sum" by the symbol Σ. Constructions like your ∞^∑(n=1) a_n are particularly ugly and hard to read, and do not make sense when parsed according to standard rules for reading mathematical expressions.
     
  8. Jan 9, 2017 #7
    Thanks, I'll keep that in mind if something else comes up. The next time I post, I'll try and use LaTeX for the equations; although, I need to learn how to use it first.
     
  9. Jan 9, 2017 #8

    Ray Vickson

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    There are tutorials available in this Forum, but perhaps the easiest way is to look at someLaTeX statements of others. For example, you can write ##\pi^3/2 + \sum_{n=1}^{\infty} (-1)^n 4 / n^2 \cos(nx)## or ##\pi^3/2 + \sum_{n=1}^{\infty} (-1)^n 4 / n^2 \cos nx## for an in-line expression and
    $$\sum_{n=1}^{\infty} (-1)^n \frac{4}{n^2} \cos (nx) \; \text{or} \; \sum_{n=1}^{\infty} (-1)^n \frac{4}{n^2} \cos nx$$
    for a displayed expression.

    To see the constructions used, just double right-click on the image and ask for a display of math as tex. The in-line version is initiated and terminated by "# #" (with space between the two #s removed), so # # some formula # #. The displayed version uses "$ $" instead (again, with no spaces between the two $s). Note in particular the use of "\cos" instead of "cos"; try it both ways and see why.
     
  10. Jan 9, 2017 #9
    I see. I'll have a play around with it and see what I can do. Thank you
     
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