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!

Sequence limits?

  1. Sep 9, 2009 #1
    Let lim n →∞ XnYn = 0. Is it true that Limn →∞ Xn= 0
    or Limn →∞Yn = 0 (or both)?
     
  2. jcsd
  3. Sep 9, 2009 #2
    lim n →∞ XnYn = lim n →∞ Xn * lim n →∞ Yn

    One or even both limits need to be 0 so that lim n →∞ Xn * Yn =0

    I guess you are right. :smile:
     
  4. Sep 9, 2009 #3

    VietDao29

    User Avatar
    Homework Helper

    Well, this is only true when both xn, and yn have limits. So, what if they don't? :wink: Say, what if they're oscillating?
     
  5. Sep 9, 2009 #4

    This equation is only correct if the individual limits on the right-hand side exist. The proposition in the original post is false and can be disproven with a counterexample.
     
  6. Sep 10, 2009 #5
    Yup, sorry I forgot to mention that the series must converge, in other way they would be divergent and we could not separate them.
     
  7. Sep 10, 2009 #6
    Simple counterexample. Clearly, [tex]\lim 1/n = \lim \(n \cdot 1/n^2 \) = 0[/tex] but we have that [tex]\lim n = +\infty[/tex] and [tex]\lim 1/n^2 = 0[/tex].
     
  8. Sep 10, 2009 #7
    This example satisfies the original claim that at least one of the sequences converges to 0. A counter example for this must consist of two sequences whose product vanishes but both sequence do not converge to 0.

    --Elucidus
     
  9. Sep 11, 2009 #8

    VietDao29

    User Avatar
    Homework Helper

    You've misquoted yet again, Elucidus.. :cry: :cry:

    You shouldn't forget to wear your glasses, then.. :wink:
     
  10. Sep 11, 2009 #9
    Indeed. I'm surprised. I was trying to quote this post:

    To which my original comment makes more sense. Sorry for any confusion.

    --Elucidus
     
  11. Sep 11, 2009 #10
    Let [itex]x_n [/itex] be alternately 1 and 0, let [itex]y_n [/itex] be alternately 0 and 1.
    Then [itex]x_n y_n = 0 [/itex] for all n, but neither individual limit exists. In particular, neither individual limit is zero.
     
  12. Sep 11, 2009 #11

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Excellent example. However, neither [itex]x_n[/itex] nor [itex]y_n[/itex] converges and the O.P. finally told us.
     
  13. Sep 11, 2009 #12
    HallsofIvy I am not the O.P, mate. :biggrin: I was just adding, additional explanation of my first post. :cool:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Sequence limits?
  1. Sequences and Limits (Replies: 8)

  2. Limit of a Sequence (Replies: 18)

  3. Limits and Sequences (Replies: 8)

  4. Limit of a Sequence (Replies: 1)

Loading...