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For what values of p does the series converge?

  1. Nov 7, 2007 #1
    1. The problem statement, all variables and given/known data
    For what values of p does the series converge?


    2. Relevant equations
    Σn=2 1/(n^p)(ln n)


    3. The attempt at a solution
    So far, all that is available to me is the integral test, the comparison test and the limit comparison test.

    So using the comparison test. 1/(n^p)(ln n) < 1/n^p

    1/n^p is a p series and only converges when p>1. So it converges.

    But this doesn't feel right :(
     
  2. jcsd
  3. Nov 7, 2007 #2

    dynamicsolo

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    Homework Helper

    This is fine: the comparison of terms is valid for n > 1 (so the summation in the series would have to start at n = 2).

    In having looked at a fair number of these p-series related problems, I've come to think of the contribution of a (ln n) factor as equivalent to adding zero to the p exponent when conducting the "p-test"; the natural logarithm in such series seems to make a negligible contribution to the convergence of the series.
     
  4. Nov 7, 2007 #3
    ah i meant to add in so it converges for p>1

    its just that all the p series problems usually end up with p>1 XD
     
  5. Nov 7, 2007 #4

    dynamicsolo

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    That's largely because of the problems authors tend to choose for courses making a first pass through material on infinite series. Not all p-series one may encounter turn out that way... ;-)
     
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