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Possible Duplicate:
What does recursive cosine sequence converge to?

Consider the following sequence defined as follows:

$x_0 =1.$ $x_{n+1} = \cos(x_n)$

How do I show that $x_n$ is convergent?

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    You are doing what is sometimes called a fixed point iteration. If you can prove that the derivative (of $\cos x)$ stays below $c$ in absolute value, where $c$ is a definite number less than $1$, then there is convergence. Note that being <1 in absolute value is not enough.2012-03-01

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