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Series representation

  1. Jan 12, 2014 #1
    [itex]\sum\limits_{m=-N}^N e^{-i m c} = \frac{sin[0.5(2N+1) c]}{sin[0.5 c]}[/itex]

    I have to show the equality. But I'm absolutely dumbfounded how to even begin. I always hated series. I tried to use Euler's identity.

    [itex]e^{-i m c} = cos(mc) - i sin(mc)[/itex]

    Then I tried to sum over the 2 terms separately. But I'm not sure if this is even valid and I certainly don't get what I want. Any ideas?
     
  2. jcsd
  3. Jan 12, 2014 #2

    maajdl

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    Gold Member

    This is a geometric series.
     
  4. Jan 12, 2014 #3
    I have been told this before but I don't see how this helps me to be honest.

    ∑[itex]q^x = \frac{1 - q^{n+1}}{1 - q}[/itex]

    I'm surely overlooking something ... How do I apply this to my exp function?
     
  5. Jan 12, 2014 #4

    jbunniii

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    First, note that this formula is correct if the sum is taken from ##0## to ##n##. Your sum goes from ##-N## to ##N##, so you will have to manipulate it before you can apply the formula.

    If you don't see why your series is geometric, note that ##e^{-imc} = z^m## where ##z = e^{-ic}##.
     
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