Investigate sum

  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. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    Do you mean:
    [tex]
    \sum_{k=1}^\infty (\sqrt{k+1} - \sqrt{k})(\ln{(k+1)}-\ln{k})
    [/tex]?
     
  4. yes.. thnx.
     
  5. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    Hint: sqrt(k+1)-sqrt(k) = {sqrt(k)+sqrt(k+1)}^{-1}
    and you can put the logs together.
     
  6. Got it. Thnx matt.
     
  7. 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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