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Math Help - Power series (Complex)

  1. #1
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    Power series (Complex)

    Find the power series about the origin for the given function.

    (1) \frac{z^3}{1-z^3} , |z| < 1

    (2) z^2cosz

    I will try to attempt the first one:

    (1) \frac{1}{(1-z)^3} = (\frac{1}{1-z})'

    \frac{1}{1-z} = 1 + z + z^2 + z^3 + ....

    \frac{z}{1-z} = z + z^2 + z^3 + z^4 + ...

    \frac{z^3}{1-z} = z^3 + z^6 + z^9 + z^{12} + ...

    = \sum_{n = 1}^{\infty} z^{3n}

    Somehow, I think my reasoning is wrong. The answer is correct, but I am not sure I took the correct steps.

    I have no idea how to approach (2). Thanks for looking and any help provided!
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  2. #2
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    Quote Originally Posted by shadow_2145 View Post
    (1) \frac{z^3}{1-z^3} , |z| < 1
    \frac{1}{1-z^3} = 1 + z^3+z^6+... now multiply by z^3

    (2) z^2cosz
    z^2 \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n}}{(2n)!}
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