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Complex arithmetic/Geo Series proof

  1. Jan 27, 2010 #1
    1. The problem statement, all variables and given/known data
    Applications from multiple-slit diffraction involve sums like the following. Prove that:

    [tex]\sum^{N-1}_{n=0}[/tex] cos (nx) = [tex]\frac{sin(N(x/2))}{sin(x/2)}[/tex] * cos((N-1)x/2)

    2. Relevant equations

    According to my instructions, this should involve only algebraic manipulations


    Also there is this hint:

    Use the geometric series formula:

    [tex]\sum^{M}_{n=0}[/tex] r[tex]^{n}[/tex]= [tex]\frac{1-r^{M+1}}{1-r}[/tex]



    3. The attempt at a solution
    Once again, I apologize for the poor use of Latex. I hope the equation is easily understood. This problem comes from the complex arithmetic section of my book. I originally followed a lot of different trig identity paths until I re-read the part about using only algebraic manipulations.

    If possible, I would like a small shove past the hint. I'm not asking for a complete proof here, only a little more light in the right direction. Thanks for taking the time to read.
     
  2. jcsd
  3. Jan 27, 2010 #2

    Dick

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    Science Advisor
    Homework Helper

    cos(nx) is the real part of exp(inx). So the sum of cos(nx) is the real part of the sum of exp(ix)^n. The exponential form is a geometric series.
     
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