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Alternating series where the limit goes to zero BUT it diverges?

  1. Aug 5, 2010 #1
    1. The problem statement, all variables and given/known data

    My prof gave us an extra credit opportunity for a few extra points on the final exam(tomorrow).

    He told us to go find an example of an alternating series that is decreasing, its limit->0, and it diverges. So far I haven't seen any examples, plus I have sat around some tonight instead of studying for the final trying to figure this one out.






    2. Relevant equations

    Alternating series test. But I am bad at Latex so Ill spare everyone.

    3. The attempt at a solution

    I'm not sure if im right on this one, but how about an alternating p-series with p=1? If you take the limit of 1/n as n->infinity the series goes to zero. Its also decreasing AND according to the p-series test, if p<=1 then it diverges. Is this a good answer or..?

    We have actually used this on before in class...so he may be looking for a more original alternating series.

    By the way, I have to actually make up my own problem and solve it, but that will be the easy part.
     
  2. jcsd
  3. Aug 5, 2010 #2

    Char. Limit

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    But the p-series applies to functions of the sort...

    [tex]\sum_{n=1}^\infty \frac{1}{n^p}[/tex]

    Which is non-alternating. Also, the alternating harmonic series you describe...

    [tex]\sum_{n=1}^\infty \frac{\left(-1\right)^{n-1}}{n}[/tex]

    Does converge, as do all alternating series that are absolutely decreasing.
     
  4. Aug 5, 2010 #3
    so...no such series exist? I must have misheard my professor/
     
  5. Aug 5, 2010 #4

    Char. Limit

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    Or he could be cruel. I'm pretty sure no such series exists however. If one did, the alternating series test would be invalid.
     
  6. Aug 5, 2010 #5
    A Leibniz Series is:

    1. alternating i.e. (-1)^n.
    2. lim |a| -> 0
    3. decreasing.

    any series that has this 3 conditions, converges conditionally :)
     
  7. Aug 6, 2010 #6

    HallsofIvy

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    A man after my own heart!

     
  8. Aug 6, 2010 #7
    I think the trick is to find one that's decreasing to 0, but not monotonically decreasing:

    1 - 0 + 1/2 - 0 + 1/3 - 0 + ···

    1/1 - 1/12 + 1/2 - 1/22 + 1/3 - 1/32 + ···
     
  9. Aug 6, 2010 #8

    Char. Limit

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    Maybe you could use the bernoulli numbers. Every other n, B_n is zero, so if you co-ordinated that with the (-1)^n just right...
     
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