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Math Help - Tests for nonabsolute convergence

  1. #1
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    Tests for nonabsolute convergence

    Let 0<a<1 and consider the series
    a^2 + a + a^4 + a^3 + ... +a^(2n) + a^(2n-1) + ...
    Show that the Root test applies but the ratio test does not apply.

    Thanks in advance
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  2. #2
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    Re: Tests for nonabsolute convergence

    Is there any reason we can't write this as \displaystyle \begin{align*} a + a^2 + a^3 + a^4 + \dots \end{align*} and state that it is a geometric series with \displaystyle \begin{align*} t_1 = a \end{align*} and \displaystyle \begin{align*} r = a \end{align*}, therefore the series will be convergent if \displaystyle \begin{align*} |a| < 1 \end{align*}?
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  3. #3
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    Re: Tests for nonabsolute convergence

    I don't think so. Because the question specifically asks to show that root test applies but ratio test doesn't.
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