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Fourier series convergence

  1. Jun 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Screenshot2012-06-15at45935AM.png

    This series is what dictates the graph above.

    Screenshot2012-06-15at50422AM.png



    3. The attempt at a solution

    I don't understand what's going on. If they're using the series that i pasted below then why aren't they multiply each value in the brackets by -2/pi?

    I also don't get why terms in the brackets are

    [(-2pi + 0) + f(-2pi - 0)]

    Where did they come from? Where does 1/2 come from that multiplies the brackets. very confused.
     
  2. jcsd
  3. Jun 15, 2012 #2

    LCKurtz

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    Science Advisor
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    Gold Member

    They could have. The ##\frac {-2}\pi## was just factored out.
    Your function isn't defined at the jumps, like at ##-2\pi##. To get the average value at the jumps you need the average of the right and left limits:$$
    \lim_{h\rightarrow 0+}\frac{f(-2\pi + h)+f(-2\pi -h)}{2}$$Those limits give the value of the function at ##-2\pi\ ##"coming from" the left and right. The expression ##f(-2\pi + 0)+ f(-2\pi -0)##is just a shorthand notation for the limits, and you add them and divide by 2 to get the average.
     
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