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Convergence of Series

  1. Nov 4, 2011 #1
    1. The problem statement, all variables and given/known data

    "Determine whether the following series converge:

    [itex]\sum_{n \geq 2} \frac{n^{ln (n)}}{ln(n)^{n}}[/itex]

    and

    [itex]\sum_{n \geq 2} \frac{1}{(ln(n))^{ln(n)}}[/itex]

    2. Relevant equations

    The convergence/divergence tests (EXCEPT INTEGRAL TEST):

    Ratio
    Dyadic
    Comparison
    P-test
    Cauchy Criterion
    Root Criterion
    Alternating Series Test/Leibniz Criterion
    Abel's Criterion

    3. The attempt at a solution

    My TA said it was helpful to use the Dyadic Criterion to solve series involving logs... I believe this is an exception. It made the equation really convoluted:

    [itex]\sum_{n \geq 2} \frac{2^{2k}*k*ln(2)}{(k*ln(2))^{2^{k}}}[/itex]

    I'm sure I have to use some combination of the tests, but I kind of need to be pointed in the right direction... I have no idea how to work with that series..

    Thank you!
     
  2. jcsd
  3. Nov 4, 2011 #2
    Try the root test; C=lim{n->inf} sup n^(ln(n)/n)/ln(n). Then Let u=ln(n) and substitute this into the root test. Answer should converge to C=0. So the series converges absolutely.
     
  4. Nov 4, 2011 #3
    Ok.. I tried to root test, but I'm not sure how I can take the limsup of what I get:

    n^(u/n)/u
     
  5. Nov 4, 2011 #4
    u=ln(n) → eu2e-u/u and so you get convergence to 0.
     
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