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Math Help - power series derivative

  1. #1
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    power series derivative

    Hi, can anyone please help me with this question?

    Consider the powerseries:
    f(z)=\displaystyle \sum_{n=0}^{\infty} a_n z^n
    with the property that
    f(\frac{1}{a})=\frac{1}{a^3} \forall n\in\mathbb N
    Prove that
    f'(0)=0

    I had hint to use Uniqueness Theorem of Power Series, but I still have no idea how to do it. Please help me, thanks a lot.
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  2. #2
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    You mean f(\frac{1}{n})= \frac{1}{n^3}\forall n\in \mathbb N.

    For the power series \displaystyle f(z)= \sum_{n=0}^\infty a_nz^n, f'(z)= \displaystyle\sum_{n=0}^\infty} na_nz^{n-1} and, in particular, f'(0)= a_1. You are really asked to prove that a_1= 0.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by tsang View Post
    Hi, can anyone please help me with this question?

    Consider the powerseries:
    f(z)=\displaystyle \sum_{n=0}^{\infty} a_n z^n
    with the property that

    f(\frac{1}{a})=\frac{1}{a^3} \forall n\in\mathbb N

    Prove that

    f'(0)=0

    I had hint to use Uniqueness Theorem of Power Series, but I still have no idea how to do it. Please help me, thanks a lot.
    If f(\frac{1}{a})= \frac{1}{a^{3}}\ , \ \forall a \in \mathbb{N} that means that is...

    \displaystyle \sum_{n=0}^{\infty} \frac{a_{n}}{a^{n}} = \frac{1}{a^{3}}\ , \ \forall a \in \mathbb{N} (1)

    ... so that is...

    a_{n}=\left\{\begin{array}{ll}1 ,\,\, n=3\\{}\\0 ,\,\, n \ne 3\end{array}\right. (2)

    ... and from (2) You conclude that is f^{'}(0)=0...

    Kind regards

    \chi \sigma
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