1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

    User Avatar
    Gold Member

    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

    User Avatar
    Gold Member

    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


    User Avatar
    Science Advisor

    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

    User Avatar
    Gold Member

    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...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook