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Math Help - Dirichlet kernel

  1. #1
    Super Member Deadstar's Avatar
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    Dirichlet kernel

    How would you show that \sum_{k=-n}^n e^{ikx} =\frac{\sin\left(\left(n+\frac{1}{2}\right)x\right  )}{\sin(x/2)}

    WITHOUT using the method shown here...
    Dirichlet kernel - Wikipedia, the free encyclopedia

    Alternatively how would you prove that

    \sum_{k=-n}^n r^k=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Deadstar View Post
    How would you show that \sum_{k=-n}^n e^{ikx} =\frac{\sin\left(\left(n+\frac{1}{2}\right)x\right  )}{\sin(x/2)}

    WITHOUT using the method shown here...
    Dirichlet kernel - Wikipedia, the free encyclopedia

    Alternatively how would you prove that

    \sum_{k=-n}^n r^k=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}?
    The second is trivial (and hence so is the first), multiply by r^n and you have \sum_{k=1}^{2n} r^k which is well known, but this is what the Wikipedia article does.

    CB
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