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Math Help - Power Series Convergence Radius

  1. #1
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    Power Series Convergence Radius

    Find the radius of convergence of R of the following power series.

    \sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}

    I don't know what to do with the x^{2k} term.

    Thanks
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by tbyou87 View Post
    Find the radius of convergence of R of the following power series.

    \sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}

    I don't know what to do with the x^{2k} term.

    Thanks
    did you try the root test?
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  3. #3
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    Quote Originally Posted by tbyou87 View Post
    Find the radius of convergence of R of the following power series.

    \sum_{k=0}^{\infty}\frac{k2^k}{3^k}x^{2k}

    I don't know what to do with the x^{2k} term.

    Thanks
    Use the root test,
    \left| \frac{k2^k}{3^k}x^{2k} \right|^{1/k} = \frac{2}{3}k^{1/k}x^2 the limit of k^{1/k}\to 1 thus:
    |a_kx^{2k}|^{1/k} \to \frac{2}{3}x^2.
    We require that,
    \frac{2}{3}x^2 < 1 \implies x^2 < \frac{3}{2}.
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