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Math Help - Alternating Series

  1. #1
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    Alternating Series

    Hi everyone,

    I have the Series (-4) ^ K / K^2 and have to find if its absolutely convergent, conditionally convergent or divergent. I know the the generic form of alternating series is (-1)^k or (-1)^ k+1 so i wasnt sure how to go about this one. Also the Series cos(pie)k / k..Since its bounded between -1 and 1 can i just make l Uk l =1/k...?

    Thanks for your time and help!
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  2. #2
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    Quote Originally Posted by nikie1o2 View Post
    Hi everyone,

    I have the Series (-4) ^ K / K^2 and have to find if its absolutely convergent, conditionally convergent or divergent. I know the the generic form of alternating series is (-1)^k or (-1)^ k+1 so i wasnt sure how to go about this one. Also the Series cos(pie)k / k..Since its bounded between -1 and 1 can i just make l Uk l =1/k...?

    Thanks for your time and help!
    For the first series
    \sum_{k=1}^{\infty} \frac{(-1)^k 4^k}{k^2}

    Since
    \lim_{k \to \infty} \frac{4^k}{k^2} \to \infty

    your series will diverge. For the second, considering

     \sum_{k=1}^{\infty} \frac{ \cos k \pi }{k} = \sum_{k=1}^{\infty} \frac{ (-1)^k }{k}

    Since
     \sum_{k=1}^{\infty} \frac{ 1}{k} diverges (it's harmonic) you should use the alternating series test.
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  3. #3
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  4. #4
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    Quote Originally Posted by danny arrigo View Post
    For the first series
    \sum_{k=1}^{\infty} \frac{(-1)^k 4^k}{k^2}

    Since
    \lim_{k \to \infty} \frac{4^k}{k^2} \to \infty

    your series will diverge. For the second, considering

     \sum_{k=1}^{\infty} \frac{ \cos k \pi }{k} = \sum_{k=1}^{\infty} \frac{ (-1)^k }{k}

    Since
     \sum_{k=1}^{\infty} \frac{ 1}{k} diverges (it's harmonic) you should use the alternating series test.


    Thanks you. I also am stuck on  \sum_{k=2}^{\infty} \frac{(-1)^k}{klnk} ...
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  5. #5
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    The same, Leibniz test.

    Prove first that a_k=\frac1{k\ln k} is a strictly decreasing sequence for all k>\frac1e.
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