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How to test this serie for convergence?

  1. May 19, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm trying to determine if Ʃ 1/(3^ln(n)) converges.

    2. Relevant equations



    3. The attempt at a solution

    The preliminary test isn't of any help since lim n→∞ an = 0.

    I tried the integral test but I couldn't integrate the function, and I don't think it's the best way to proceed. I couldn't do anything with the ratio test either, since I don't know how to simplify the 1/(3^ln(n+1)) term.
     
  2. jcsd
  3. May 19, 2012 #2

    micromass

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    Do you know what [itex]e^{ln(x)}[/itex] is??
     
  4. May 19, 2012 #3
    I know it equals x.

    So the sum of (3^ln(x))^-1 is smaller than the sum of (x)^-1. I guess I could use a comparison test here, thanks.
     
  5. May 19, 2012 #4

    sharks

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    That an ingenious way of solving this problem.

    I might have solved it using micromass' hint. Try letting: [itex]y = 3^{\ln n}[/itex]
     
  6. May 19, 2012 #5
    y = 3^ln(x)

    ln(y) = ln[3^ln(x)]

    ln(y) = ln(x) ln(3)

    I'm stuck here :confused:
     
  7. May 19, 2012 #6
    3ln x = eln(3ln x) = e(ln x)(ln 3) = (eln x)ln 3 = xln 3 (and here we have a nice log/exponential identity!)

    So 1/3ln n = 1/nln 3
     
  8. May 20, 2012 #7
    Thanks, that's very clever!
     
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