A new method for determining the limits of recursion relations

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This is meant as a note. I will definitely expand this article later.

I hope this method has not been discovered before :(.

Say x_n=f(x_{n-1}) is a recursion relation. Then \lim\limits_{n\to\infty} x_n is the intersection point of y=f(x) and x=f(y).

For example, if we have x_n=\sqrt{5 x_{n-1}+6}, then \lim\limits_{n\to\infty} x_n is the y-coordinate of the intersection of y=\sqrt{5x+6} and x=\sqrt{6y+5}.

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Graduate student

One thought on “A new method for determining the limits of recursion relations”

  1. Nice post, but it will be more illuminating if you can give an example where solving the simultaneous equations $x=f(y)$ and $y=f(x)$ is easier than solving $x=f(x)$

    Reply

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