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Fourier Series Problem

  1. Oct 22, 2014 #1
    1. The problem statement, all variables and given/known data
    Define ##f : [−π, π) → \mathbb R ## by
    ##f(x)## = ##−1## if ##− π ≤ x < 0##, ##1## if ##0 ≤ x < π.##
    Show that the Fourier series of f is given by
    ##\frac{4}{π} \sum_{n=0}^\infty \frac{1}{(2k+1)} . sin(2k+1)x##


    2. Relevant equations

    The Fourier series for ##f## on the interval ##[−π, π)## is given by:

    ##\frac{a0}{2} + \sum_{n=0}^\infty ancos(nx) + \sum_{n=0}^\infty bnsin(nx)##
    (Not quite sure why the LaTex isn't working here, I'm new at this.)
    Where a0/2, an, bn are the Fourier coefficients of ##f##.

    3. The attempt at a solution

    I have attempted to find the Fourier coefficients, however I don't think that they're correct.

    I have found
    a0/2 = 1
    an = 0.
    bn = ##\frac {2}{πn} -2\frac {(-1)^n}{n}##

    Could somebody tell me if these are correct? If not, I'll post up how I reached those answers.
     
  2. jcsd
  3. Oct 22, 2014 #2

    Dick

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    I think you lost a ##\pi## in the second term of your expression for ##b_n##.
     
  4. Oct 23, 2014 #3

    vela

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    ##a_0## isn't correct either.
     
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