Sequence Convergence & Fixed Point Theorem

[f00lishroy]

Homework Statement

Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by p_n= g(p_n-1) where n≥1

a) Determine the values of p₀ \in [1,2] for which the sequence {p_n} from 0 to infinity converges.

b) For the cases where {p_n} converges (if any), what is the rate of convergence?

Homework Equations

http://en.wikipedia.org/wiki/Fixed-point_theorem

Fixed Point Theorem

The Attempt at a Solution

For part a, my answer is that ANY point p0 between 1 and 2 will converge, because the sequence satisfies the fixed point theorem.


g(x) exists on [1,2] and is continuous
g'(x) = (2/3)(1-2/(x^3)) exists and is continuous on [1,2]

There is a positive constant k<1 for which |g'(x)|≤k

By plotting g'(x), i found k = 2/3

Therefore, the fixed point theorem is satisfied, and so should the answer be "any value of p0 from 1 to 2 will cause the sequence to converge"?

For part b, I am not sure how to find the rate of convergence. I heard that you have to take the taylor series expansion, but I am not sure. Any help? Thanks

 

[haruspex]

Yes, that shows it converges throughout the interval.
For the rate, let p be the fixed point and x be small. If the rate of convergence is a then g(p+x) is approximately p + ax, i.e. the error gets multiplied by factor a at each step. Take the Taylor expansion of g at p, just the first two terms, and see what that simplifies to.