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Convergence of a Series

  1. Mar 19, 2008 #1
    1. The problem statement, all variables and given/known data

    Convergent/Divergent?

    Problem 1: infinite series, sigma(n=1) [1/((n^2)-n)
    This one is clearly convergent, however it is undefined at n=1.
    Is the whole problem undefined? Or is it convergent from (n=2)?


    Problem 2: infinite series, sigma(n=1) [ln(n)/n]
    This is divergent. It works with the Limit Comparison Test, but for some reason, I'm thinking I have to start by using sigma(n=2) [ln(n+1)/(n+1)] because, otherwise the first term would be zero. Can the test for a positive series include zero?


    2. Relevant equations

    Problem 1: Sn=(1/0)+(1/3)+(1/8)+...
    Sn - undefined

    Problem 2: Sn=(0/1)+(ln2/2)+(ln3/3)+...
    (positive series?)

    3. The attempt at a solution

    Problem 1: Sn undefined => Series also undefined?

    Problem 2: LCT, use b=(1/n), lim(n..inf) (a/b) = lim(n..inf) [(lnn/n)/(1/n)] = lim(n..inf) ln(n) = inf , limit > 0 , sigma (1/n) divergent, so sigma (lnn/n) divergent too.
    Series divergent
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 19, 2008 #2
    The convergance of a series only depends on what happens at the end.
     
  4. Mar 19, 2008 #3

    Tom Mattson

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    Yes, it's undefined. I think whoever wrote the problem botched this. You should ask your prof for clarification.

    The series does diverge, but you didn't prove it. In order for LCT to tell you anything, the limit has to come out to a finite, positive number. You won't get that here. Use the Integral Test instead.
     
  5. Mar 19, 2008 #4
    Thank you. Much appreciated.

    What about the term "positive series"? The convergence tests call for positive series, but problem 2 has a_1 = 0. I can use a_n+1 for the proof, but do I have to?
     
  6. Mar 19, 2008 #5
    (...pardon me)

    Is zero a positive number?
     
  7. Mar 19, 2008 #6

    Tom Mattson

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    It really doesn't matter, because as foxjwill has pointed out, the convergence or divergence of a series only depends on what happens for large n. Deleting a finite number of terms from a series doesn't affect the convergence or divergence at all.
     
  8. Mar 19, 2008 #7

    Tom Mattson

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    No, it isn't.
     
  9. Mar 19, 2008 #8
    It's neither positive or negative.
     
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