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Infinite alternating series with repeated logarithms

  1. Apr 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Calculate the sum of the following series:
    [tex]\sum_{i=2}^{\infty}(-1)^i \cdot \lg ^{(i)} n[/tex]

    Where (i) as a super-script signifies number of times lg was operated i.e. [tex]\lg ^{(3)} n = (\lg (\lg (\lg n))) [/tex], and n is a natural number.

    2. Relevant equations



    3. The attempt at a solution

    I proved by Leibniz test that this series converges. But I don't know how to find the number to which it converges to.

    Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 8, 2009 #2
    I really don't know how to help you... but since you know it converges you can use the numerical estimate to see if it looks like any famous number?

    I hope someone helps I'm really interested in this!
     
  4. Apr 8, 2009 #3

    Dick

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    If you try doing a numerical estimate it will blow up. Which is informative. Eventually ln^(k)(n)<1. Then ln^(k+1)(n)<0. Now what's ln^(k+2)(n)?? The series isn't even well defined much less convergent.
     
  5. Apr 8, 2009 #4
    You are right.
    The series isn't well defined so I cancel this question. I apologize.
     
  6. Apr 8, 2009 #5
    Oh that makes me feel a little better. I could not even see how you had it converging.

    :tongue:
     
  7. Apr 8, 2009 #6

    Dick

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    No need for apologies!! It's a perfectly valid question. It makes a good 'think about it' exercise. It does look at first glance like it ought to be a good candidate for an alternating series test.
     
    Last edited: Apr 8, 2009
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