Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Convergence doubt

  1. Oct 22, 2014 #1

    phoenix95

    User Avatar
    Gold Member

    Hello people,
    In case if I am typing the question in the wrong forum please redirect me.
    OK, here it goes...... I have this stupid question:
    Suppose we have a sequence 1, 1, 1, 1, 1, 1........ It converges to '1'.
    Consider 1, 0, 1, 0, 1, 0........ it diverges right?
    What about a sequence 1, 1, 0, 1, 1, 0..... ?? Does it converge?
    What about 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0,........?
    What about any similar random sequence? does it converge to the most repeated value?

    Thanks for any help in advance.
     
  2. jcsd
  3. Oct 22, 2014 #2
    Any time the limit superior of a sequence is not equal to the limit inferior, the limit will not exist.
     
  4. Oct 22, 2014 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Convergence by definition means getting close to a single value. So, no sequence with infinitely many 0's and 1's can converge.
     
  5. Oct 22, 2014 #4

    phoenix95

    User Avatar
    Gold Member

    I don't know what superior and inferior limits are, will you please explain?

    How about this?
    An infinite sequence with first few terms being 'a', and the rest all being 'b', like
    a, a, a, a, a, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b, b.................... ??
    Will it converge??
     
  6. Oct 22, 2014 #5

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That sequence converges to b. Convergence depends on the long-term behaviour of a sequence. The initial terms do not affect this.
     
  7. Oct 22, 2014 #6

    phoenix95

    User Avatar
    Gold Member

    Well then, what are these superior and inferior limits?
     
  8. Oct 22, 2014 #7

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You can google for this: limsup and liminf. But, unless you are studying real analysis seriously, I think these might be a bit difficult to digest.

    The formal definition of convergence (again you can google this) is really the beginining of "rigorous" mathematics or "analysis". Most people find they have to work quite hard to get to grips with these concepts.

    http://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_Definition
     
  9. Oct 22, 2014 #8

    phoenix95

    User Avatar
    Gold Member

    Ok, guys I got it.... According to the problem, limsup=1 and liminf=0 isn't it? That's why the sequence converges...?? Well, thanks GFauxPas and PeroK....
     
  10. Oct 22, 2014 #9

    Mark44

    Staff: Mentor

    That's why the sequence doesn't converge (i.e., it diverges).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Convergence doubt
  1. A doubt (Replies: 3)

  2. Mathcad Doubt (Replies: 1)

  3. Basic math doubt (Replies: 8)

  4. Bounded sequence doubt (Replies: 1)

Loading...