- #1
f00lishroy
- 6
- 0
Homework Statement
Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by pn= g(pn-1) where n≥1
a) Determine the values of p0 [itex]\in[/itex] [1,2] for which the sequence {pn} from 0 to infinity converges.
b) For the cases where {pn} 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