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Homework Help: Convergence of 1/(n*n^(1/n))

  1. Oct 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Test for convergence the series:
    [tex] a_[n] = \frac{1}{n*n^{\frac{1}{n}}} [/tex]

    2. Relevant equations
    Various Sequence Convergence Tests

    3. The attempt at a solution
    So far I've tried both a normal comparison and limit comparison test with n^2. The normal one seemed fine until the end. Here was my logic:

    For n greater than 1 (its just less than, not equal)

    [tex] n^{\frac{1}{n}} \le n [/tex]

    [tex] \frac{1}{n^{\frac{1}{n}}} \ge \frac{1}{n} [/tex]

    [tex] \frac{1}{n*n^{\frac{1}{n}}} \ge \frac{1}{n^2}} [/tex]

    But that doesn't work because it just proves that for every term, this sequence is greater than the p-series for n^2.

    For the limit comparison test I don't get an actual limit, so I can't use it.

    If anyone has any suggestions for which test to use, or what series to compare it to, I would be most grateful.
  2. jcsd
  3. Oct 5, 2009 #2
    Would [tex] n^{\frac{1}{n}} \le \log n[/tex] work?
  4. Oct 5, 2009 #3
    uh, I'm pretty sure that your inequality is backwards
  5. Oct 5, 2009 #4
    I'm pretty sure I'm right for "large" n. And on further reflection, n^(1/n) < 2 is even better.
  6. Oct 5, 2009 #5
    thanks, the comparison with 2 works perfectly
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