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Math Help - Match each of the following with the correct statement.

  1. #1
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    Match each of the following with the correct statement.

    Match each of the following with the correct statement.
    C stands for Convergent, D stands for Divergent.

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    5.
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  2. #2
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    Quote Originally Posted by ewkimchi View Post
    Match each of the following with the correct statement.
    C stands for Convergent, D stands for Divergent.

    1.
    2.
    3.
    4.
    5.
    1. \frac{8 + 9^n}{9^n} = \frac{8}{9^n} + 1

    So \lim_{n \to \infty}\frac{8 + 9^n}{9^n} = \lim_{n \to \infty}\left(\frac{8}{9^n} + 1\right)

     = 1

     > 0.

    So the series is divergent.
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    Quote Originally Posted by ewkimchi View Post
    Match each of the following with the correct statement.
    C stands for Convergent, D stands for Divergent.

    1.
    2.
    3.
    4.
    5.

    4. \sum_{n = 1}^{\infty}n\,e^{-n^2} \leq \int_1^{\infty}x\,e^{-x^2}\,dx

     = -\frac{1}{2}\int_1^{\infty}{-2x\,e^{-x^2}\,dx}

     = \lim_{\varepsilon \to \infty}-\frac{1}{2}\left[e^{-x^2}\right]_1^{\varepsilon}

     =\lim_{\varepsilon \to \infty}-\frac{1}{2}\left[e^{-\varepsilon^2} - e^{-1}\right]

     = -\frac{1}{2}\left[0 - e^{-1}\right]

     = \frac{1}{2}\,e^{-1}.


    By the integral test, the series converges.
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