1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Using a Fourier Cosine Series to evaluate a sum

  1. Mar 3, 2014 #1
    1. The problem statement, all variables and given/known data

    a) Show that the Fourier Cosine Series of [itex]f(x)=x,\quad 0\leq x<L[/itex] is
    [tex]x ~ \frac{L}{2}-\frac{4 L}{\pi ^2}\left[\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5 \pi x}{L}\right)}{5^2}+\dots\right][/tex]

    b) use the above series to evaluate the sum[tex]1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...[/tex]

    2. Relevant equations

    Fourier Cosine Series General form

    3. The attempt at a solution

    So I have done part a, but I am lost on how to do part b. I don't understand how to get the even terms? Perhaps I need to differentiate the series, which would pull an n out, making the n's even.

    To differentiate term by term a cosine series I need f'(x) to be piecewise continuous. Which I think it is.
     
    Last edited: Mar 3, 2014
  2. jcsd
  3. Mar 3, 2014 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Fixed LaTeX:

     
  4. Mar 3, 2014 #3
    thanks. I did too. I had to quickly reinstall LaTeXiT from macports :smile:
     
  5. Mar 3, 2014 #4
    but my fourier series only has the odd-n terms?
     
  6. Mar 3, 2014 #5

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    I'll assume the Fourier expansion is correct.

    What you need to do is to choose ##x## (and perhaps ##L##) wisely in order to extract an interesting series.

    Now, also note the following:

    Take

    [tex]S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + ...[/tex]

    then

    [tex]S = \left(1 + \frac{1}{3^2} + \frac{1}{5^2} + .... \right) + \left(\frac{1}{2^2} + \frac{1}{4^2} + ...\right)[/tex]

    But

    [tex]\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + ...= \frac{1}{4}\left(1 + \frac{1}{2^2}+ \frac{1}{3^2} + ...\right)= \frac{S}{4}[/tex]

    So differentiating is unnecessary.
     
  7. Mar 3, 2014 #6
    ok my initial guess was wrong obviously. becuase it would not make the n's even, it would make them not squared...
     
  8. Mar 3, 2014 #7

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    First think about, what you get, setting [itex]x=L[/itex]. Then you can think further about how to get the given series from this!
     
  9. Mar 3, 2014 #8
    ok. I don't really have time to show my work, but I ended up with π^2/6. Is this the correct answer?

    the trick I didn't get was that part with the s/4
     
  10. Mar 3, 2014 #9

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    You have the correct answer!

    http://en.wikipedia.org/wiki/Basel_problem
     
  11. Mar 3, 2014 #10

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Actually, this is the Fourier series of the even function ##f(x) = |x|, -L \leq x \leq L##, extended to the whole real line as a periodic function of period ##2L##. Basically, the person posing the question needs to specify ##f(x)## outside the desired interval ##[0,L]##. If that is not done correctly the Fourier series won't be pure 'cosine' and/or might not be continuous at ##\pm \, L##, leading to a mis-match between f and the sum of the series at those points.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Using a Fourier Cosine Series to evaluate a sum
Loading...