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Math Help - Radius of Convergence of Power Series

  1. #1
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    Question Radius of Convergence of Power Series

    Hello,

    I need a little bit help with the following power series. The exercise is to calculate the radius of convergence.

     \sum\limits_{n=1}^{\infty} (\mathrm{ln}(n))^{n} x^{n}

    I know the criterion of Euler and the stronger criterion of Hadamard.

    Euler:

    r= \frac{\mid a_{n} \mid}{ \mid a_{n+1} \mid }

    Hadamard:

     r= \frac{1}{\mathrm{limsup} ~ \sqrt[n]{\mid a_{n} \mid} }

    and of course a_{n} is

    a_{n} = (\mathrm{ln(n)})^{n}

    My problem now, is that I don't know how to calculate this expressions with this logarithm

    Thanks for help
    Last edited by Besserwisser; January 26th 2010 at 01:25 AM.
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  2. #2
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    Quote Originally Posted by Besserwisser View Post
    Hello,

    I need a little bit help with the following power series. The exercise is to calculate the radius of convergence.

     \sum\limits_{n=1}^{\infty} (\mathrm{ln})^{n} x^{n}

    I know the criterion of Euler and the stronger criterion of Hadamard.

    Euler:

    r= \frac{\mid a_{n} \mid}{ \mid a_{n+1} \mid }

    Hadamard:

     r= \frac{1}{\mathrm{limsup} ~ \sqrt[n]{\mid a_{n} \mid} }

    and of course a_{n} is

    a_{n} = \mathrm{ln}^{n}

    My problem now, is that I don't know how to calculate this expressions with this logarithm

    Thanks for help
    Is it \sum_{n = 1}^{\infty}(\ln{x})^nx^n?
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  3. #3
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    Quote Originally Posted by Prove It View Post
    Is it \sum_{n = 1}^{\infty}(\ln{x})^nx^n?
    I made typing errors. Sorry. The Power Seris is

    \sum_{n = 1}^{\infty}(\mathrm{ln}(n))^{n} x^{n}

    and

    a_{n} = (\mathrm{ln}(n))^{n}
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  4. #4
    MHF Contributor chisigma's Avatar
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    What is the neccesary condition for the convergence of an 'infinite sum'?...

    Kind regards

    \chi \sigma
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  5. #5
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    Quote Originally Posted by Prove It View Post
    Is it \sum_{n = 1}^{\infty}(\ln{x})^nx^n?
    But that would not be a power series.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Besserwisser View Post
    Hello,

    I need a little bit help with the following power series. The exercise is to calculate the radius of convergence.

     \sum\limits_{n=1}^{\infty} (\mathrm{ln}(n))^{n} x^{n}

    I know the criterion of Euler and the stronger criterion of Hadamard.

    Euler:

    r= \frac{\mid a_{n} \mid}{ \mid a_{n+1} \mid }

    Hadamard:

     r= \frac{1}{\mathrm{limsup} ~ \sqrt[n]{\mid a_{n} \mid} }

    and of course a_{n} is

    a_{n} = (\mathrm{ln(n)})^{n}

    My problem now, is that I don't know how to calculate this expressions with this logarithm

    Thanks for help
    For it to converge you clearly need \ln^n(n)\cdot x^n=\left(x\ln(n)\right)^n\to0. For this to happen you need only prove that x\ln(n) goes to zero for some x..but...there is a problem with that.
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