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Epsilon proof and recursive sequences

  1. Oct 15, 2012 #1

    I am wondering how one would go about an ε, N proof for a recursively defined sequence. Can anyone direct me to some reading or would like to provide insights of their own? This isn't for a homework problem... just general curiosity which I could not satisfy via search!

    Thank you.
  2. jcsd
  3. Oct 16, 2012 #2


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  4. Oct 16, 2012 #3
    Depending on the sequence, you may be able to solve for it explicitly and then take the limit as usual. What sequence are you working with, exactly?
  5. Oct 16, 2012 #4
    I'm not working on any sequence in particular, but I started wondering about it while doing something with infinitely nested radicals. I've proven the limit, convergence, etc., of recursive sequences, including nested radicals. I'm wondering if there is a way of doing a traditional epsilon proof using the definition of a convergent sequence. How do you go about finding an n>N such that |a_n - L | < ε? This confuses me because a_n is given recursively...

    Thank you for the links on cobwebbing! That looks interesting and I have never heard of it before.
  6. Oct 16, 2012 #5
    To maybe clarify a bit: I am suggesting that the limit value is already known (or at least the suspected value). Given an ε>0, how do I find N such that |a_n - L | < ε whenever n>N.

    For instance, if given the sequence x_1 = 1 and x_(n+1)= sqrt(1+x_n)... yielding sqrt(1+sqrt(1+sqrt(1+...))) which has the limit, if I remember correctly, value being the golden ratio.
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