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Thread: determine if it's converge

  1. #1
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    determine if it's converge

    $\displaystyle \sum (ln k)^2/k$

    use the root test or the ration test
    Last edited by wopashui; Mar 28th 2010 at 08:55 PM.
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    Try the integral test
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  3. #3
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    Quote Originally Posted by wopashui View Post
    $\displaystyle \sum (ln k)^2/k$

    use the root test or the ration test
    see attachment for solution:series converges
    Attached Thumbnails Attached Thumbnails determine if it's converge-sol.bmp  
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  4. #4
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    Quote Originally Posted by wopashui View Post
    $\displaystyle \sum (ln k)^2/k$

    use the root test or the ration test
    What values does $\displaystyle k$ go between?

    Is it $\displaystyle \sum_{k = 1}^{\infty}\frac{(\ln{k})^2}{k}$?
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  5. #5
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    Quote Originally Posted by wopashui View Post
    $\displaystyle \sum (ln k)^2/k$

    use the root test or the ration test
    Before using blindly any "test", simply look at what the summand looks like. It is positive. The logarithm grows slowly to $\displaystyle \infty$, hence $\displaystyle \frac{(\ln k)^2}{k}$ is a bit larger than $\displaystyle \frac{1}{k}$. But... wait! we know that $\displaystyle \sum_k \frac{1}{k}$ diverges! So we just have to write $\displaystyle \frac{(\ln k)^2}{k}\geq \frac{1}{k}$ when $\displaystyle k\geq 3$ to conclude: the series diverges.
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