Results 1 to 12 of 12

Math Help - Need confirmation.

  1. #1
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Need confirmation.

    I answered a question but I don't know if my answer is correct so I would appreciate if someone can confirm it. Thank you.
    By the way I did some mistake when I wrote it. its an+1-a​n
    Attached Thumbnails Attached Thumbnails Need confirmation.-2.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,672
    Thanks
    1074

    Re: Need confirmation.

    suppose

    a_n=(-1)^n

    then

    \forall n (a_{n+1}+a_n)=0

    certainly a_n doesn't converge. It is however bounded. I don't think you can play the same trick if {a} is unbounded. I'd have to think about it.
    Last edited by romsek; December 13th 2013 at 12:29 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Need confirmation.

    Hmmm, I didn't think about that, I thought if it didn't work I had to come to a contradiction algebrically from the definition of CAUCHY or something .. So in this case I have to give a counter-example ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,672
    Thanks
    1074

    Re: Need confirmation.

    Well just answer the questions.

    Does this mean a_n converges to any point? No, we showed that.
    Does this mean a_n is bounded? We haven't shown it but I suspect so. Show this.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,672
    Thanks
    1074

    Re: Need confirmation.

    Quote Originally Posted by romsek View Post
    Well just answer the questions.

    Does this mean a_n converges to any point? No, we showed that.
    Does this mean a_n is bounded? We haven't shown it but I suspect so. Show this.
    Looks like I'm wrong.

    a_n=(-1)^n log(n)

    clearly a_n is unbounded.

    a_n+a_{n+1}=(-1)^n log(n) + (-1)^{n+1} log(n+1)

    let n be even

    a_n+a_{n+1}=log(n)-log(n+1)=log(\frac{n}{n+1})\rightarrow log(1)=0

    similar if n is odd

    So you can manage to make the sum of adjacent elements converge to 0 even while the sequence itself is unbounded.
    Last edited by romsek; December 13th 2013 at 01:18 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: Need confirmation.

    Let b_n = a_{n+1} + a_n. According to the question, \lim_{n \to \infty} b_n = 0. As romsek showed, the convergence of b_n has nothing to do with the convergence of a_n. But, we can use b_n to look at properties of a_n.

    Check out \lim_{n \to \infty} (b_{n+1} - b_n). By limit laws, the limit of the difference is the difference of the limits, so long as both limits exist. Since both do, we know that \lim_{n \to \infty} (b_{n+1} - b_n) = \lim_{n \to \infty} (a_{n+2} - a_n) = 0. You can do the same for \lim_{n \to \infty} (b_{n+2} - b_{n+1}) = \lim_{n \to \infty} (a_{n+3} - a_{n+1}) = 0. So, if you define sequences c_n = a_{2n}, d_n = a_{2n+1}, perhaps you can show that both of these sequences are Cauchy (using the previous results), and hence they both converge. Since convergent sequences are bounded, the original sequence must be bounded, as well.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Need confirmation.

    Hmmmm, maybe we can take the sequence an=n , so we can say that an+1-an converges to 1? but an isn't bounded? Is this possible? i mean i know infinite minus infinite in limits is a problem but the distance between them should be 1 and it shouldn't be a problem right?(I got confused , this is completely wrong
    Last edited by davidciprut; December 13th 2013 at 01:37 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: Need confirmation.

    Quote Originally Posted by SlipEternal View Post
    ... perhaps you can show that both of these sequences are Cauchy (using the previous results), and hence they both converge.
    As romsek showed in post #5, you obviously cannot show that c_n,d_n are convergent. Oh well. Disproof by counterexample is always a good lesson!
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Need confirmation.

    Why are you looking at the sum? it says the difference between them converges to 0
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Need confirmation.

    So when I posted I wrote the sum between them but I corrected it by writing a note on the post, did you guys read it ?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Need confirmation.

    So this is the correct one , although I corrected myself when I submitted the thread, sorry for confusing
    Attached Thumbnails Attached Thumbnails Need confirmation.-corrected.png  
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,672
    Thanks
    1074

    Re: Need confirmation.

    say what? This is an entirely different problem.

    In this case it's just a Cauchy sequence that clearly converges to a point and is thus bounded.

    This proof must be on the web in a dozen places or so. Here.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Confirmation / Help
    Posted in the Calculus Forum
    Replies: 7
    Last Post: May 9th 2013, 10:43 AM
  2. Just need confirmation
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: March 6th 2011, 10:00 PM
  3. confirmation?
    Posted in the Algebra Forum
    Replies: 1
    Last Post: December 28th 2009, 05:40 PM
  4. need confirmation of counterexample
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 26th 2009, 07:47 PM
  5. Replies: 5
    Last Post: August 11th 2009, 06:37 AM

Search Tags


/mathhelpforum @mathhelpforum