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Math Help - Power Series

  1. #1
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    Power Series

    I'm stuck on this dam question:



    I mean the first parts are pretty simple but I don't see how it translates to the series, any help from you old chaps would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by gebauer1988 View Post
    I'm stuck on this dam question:



    I mean the first parts are pretty simple but I don't see how it translates to the series, any help from you old chaps would be greatly appreciated.
    Okay, what did you get for the power series for e^z+ e^{\omega z}+ e^{2\omega z}? Look at the real and imaginary parts of that.

    Do you see that \omega^3= 1 and so \omega^n= \omega^k where k= n (mod 3)?
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  3. #3
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    Thanks for that top notch response, now I am beginning to question whether I have my power series right you see. I have the sum of (3/n)e^((w^n)z)
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  4. #4
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    Quote Originally Posted by gebauer1988 View Post
    I'm stuck on this dam question:



    I mean the first parts are pretty simple but I don't see how it translates to the series, any help from you old chaps would be greatly appreciated.

    I suppose, and hope, you know that \sum\limits_{n=0}^\infty\frac{x^n}{n!}=e^x , and for n not a multiple of 3 we get 1+w^n+w^{2n}=1+e^{2\pi n\slash 3}+e^{4\pi n\slash 3}=\frac{e^{3\cdot 2\pi n\slash 3}-1}{e^{2\pi n\slash 3}-1}=0 , and 1+w^n+w^{2n}=3 , for n a multiple of 3 , so we get:

    e^z+e^{wz}+e^{w^2z}=\sum\limits_{n=0}^\infty\frac{  z^n+(wz)^n+(w^2z)^n}{n!} =\sum\limits_{n=0}^\infty \frac{z^n}{n!}\left(1+w^n+(w^n)^2\right)=3\sum\lim  its_{n=0}^\infty\frac{z^{3n}}{(3n)!}


    From the above we get \sum\limits_{n=0}^\infty\frac{8^n}{(3n)!}=\sum\lim  its_{n=0}^\infty\frac{2^{3n}}{(3n)!}=\frac{1}{3}\l  eft(e^2+e^{2w}+e^{2w^2}\right) =\frac{1}{3}\left(e^2+e^{2\displaystyle{e^{2\pi \slash 3}}}+e^{2\displaystyle{e^{4\pi\slash 3}}}\right) , and:

    e^{2e^{2\pi\slash 3}}+e^{2e^{4\pi\slash 3}}=e^{-1+\sqrt{3}i}+e^{-1-\sqrt{3}i}=2e^{-1}\cos\sqrt{3}\Longrightarrow\sum\limits_{n=0}^\in  fty\frac{8^n}{(3n)!}=\frac{e^2+2e^{-1}\cos\sqrt{3}}{3} \cong 2.4236

    You do the rest.

    Tonio
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  5. #5
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    Superb, thanks for the explanation
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