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Investigate sum

  1. May 27, 2004 #1
    [tex]
    \sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{k+1}-\ln{k})
    [/tex]

    How do I go about finding out if it's convergent or divergent ?
     
  2. jcsd
  3. May 27, 2004 #2

    arildno

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    Do you mean:
    [tex]
    \sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{(k+1)}-\ln{k})
    [/tex]?
     
  4. May 27, 2004 #3
    yes.. thnx.
     
  5. May 27, 2004 #4

    matt grime

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    Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
    and you can put the logs together.
     
  6. May 27, 2004 #5
    Got it. Thnx matt.
     
  7. May 27, 2004 #6
    after combining the logs, try to prove that
    [tex]\frac{1}{x} \geq \ln (1 + \frac{1}{x})[/tex]
    for all positive x

    edit:
    oops i missed the last reply while typing mine sorry
     
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