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Alternating series

  1. Nov 25, 2013 #1
    1. The problem statement, all variables and given/known data
    Ʃ (-1)^n [ n+ln(n) / n-ln(n)] from n = 2 to infinity.


    2. Relevant equations

    I looked at the limit first because I thought lnn was very slow function. n would go faster.

    3. The attempt at a solution

    limit n --> ∞ [ n+ln(n) / n-ln(n)] = 1 so it diverges.
    Limit is not 0 so it violates the one of the conditions? OK ? OR wrong?
    Thanks,
     
  2. jcsd
  3. Nov 25, 2013 #2

    Dick

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    Yes, it doesn't converge. If ##|x_n|## doesn't converge to 0, then ##\Sigma x_n## doesn't converge. That's true whether the series is alternating or not. You still have to prove its limit isn't zero. Just saying 'slow function' doesn't do the job.
     
  4. Nov 25, 2013 #3

    Ray Vickson

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    USE PARENTHESES! What you have written is
    [tex] \lim_{n \to \infty} n + \frac{\ln(n)}{n} - \ln(n) = 1 \, \leftarrow \text{ false}[/tex]
    Perhaps you meant
    [tex] \lim_{n \to \infty} \frac{n + \ln(n)}{n - \ln(n)} = 1,[/tex]
    which is true. What would be so hard about writing (ln(n)+n}/(ln(n)-n), or [ln(n)+n]/[ln(n)-n], if that is what you really meant?
     
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