Epsilon proof and recursive sequences

[dustbin]

Hi,

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.

 

[chiro]

Hey dustbin.

You might want to look at cobwebbing and related ideas:

http://en.wikipedia.org/wiki/Fixed_point_(mathematics)

http://en.wikipedia.org/wiki/Cobweb_plot

 

[Number Nine]

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?

 

[dustbin]

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.

 

[dustbin]

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.