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Convergent limits for sequences: please help picture terms

  1. Jan 26, 2014 #1

    939

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    A limit of a sequence is definetely convergent if:

    If for any value of K there is an N sufficiently large that an > K for n > N, OR for any value of K there is an N sufficiently large that an<±K for n > N

    My only question is what exactly are K, N, an and n? What values are they? How would they be graphed? I.e. for the sequence a(n) = 2n

    n = 2
    an = 4
    What are K and N? Are they on the horizontal or vertical axis?
     
  2. jcsd
  3. Jan 26, 2014 #2

    tiny-tim

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    hi 939! :smile:
    nooo, you mean definitely divergent :biggrin:

    essentially, it means that the sequence converges to ∞, or to -∞

    we can't use δ and ε for ∞ (because we can't get close enough to ∞ !)

    so instead of small circles round the limit, we draw large circles round the limit (∞), ie x > K (or x < -K), and we say that that large circle has to contain all an once n is large enough

    it's the same as saying that the sequence {1/an} has the limit 0 (from above, for the ∞ case) (or from below, for the -∞ case)​
     
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