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When can you not apply the alternating series test?

  1. Mar 20, 2013 #1
    1. The problem statement, all variables and given/known data

    I have a series

    Ʃ(1 to infinity) ((-1)^n*n^n)/n!



    2. Relevant equations



    3. The attempt at a solution

    apparently you cannot use the alternating series for this question, why is this? It has the (-1)^n, what else is needed to allow you to use the alternating series test?
     
  2. jcsd
  3. Mar 20, 2013 #2

    Dick

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    To apply the alternating series test you also need that the absolute value of the terms is decreasing. Yours aren't. E.g. (-1)^n*n doesn't converge.
     
  4. Mar 20, 2013 #3
    Would that not fall under one of the rules for alternating series (lim An as n->∞ must equal to 0 for the series to be convergent)? How can you determine a series to be divergent from this part of the test if you are not even able to apply it?
     
  5. Mar 20, 2013 #4

    Dick

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    If ##a_n## as a sequence does not converge to zero then the series ##\Sigma a_n## does not converge. Try that test.
     
  6. Mar 20, 2013 #5
    Is there any way to do it only using the alternating series test?

    From what i understand this is the alternating series test

    Consider Ʃ(n=1 to ∞) (-1)^(n-1)bn where bn>=0. Suppose
    I) b(n+1)<= bn eventually
    II) lim as n ->∞ bn=0
    Then Ʃ(n=1 to ∞) (-1)^(n-1)bn converges.

    It seems like what you are saying is that you cannot test for divergence using the alternating series test, only convergence?

    The reason i am asking this is beccause of this

    http://imgur.com/kTd1Lbt

    It has a choice where it can diverge and a choice where you can not apply alternating series test.

    So is it true that for an alternating series test you can test for convergence, but if it does not follow the 2 conditions above, then you must take bn (or an) and apply it to a different test to see it if diverges or converges?
     
  7. Mar 20, 2013 #6

    Dick

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    Yes, the alternating series test can only be applied to test for convergence. Failure of the alternating series test doesn't prove anything. Try another test. As I said, if ##a_n## doesn't converge to zero the series does NOT converge. That's a good first test for divergence.
     
  8. Mar 20, 2013 #7
    ok thanks for clarifying
     
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