Iteration/Root finding algorithm

[Oshada]

Homework Statement

The Attempt at a Solution

I've managed to do part a), by factorising the cubic you get when you rearrange the terms. I'm mostly stumped for part b). I know the sequence has to be contractive, otherwise it wouldn't converge. Also, how do I show it's the only solution? Thank you very much!

 

[HallsofIvy]

I'm not sure what you mean by "show it's the only solution". There are, in fact, three (complex) solutions to the equation. What you are asked to do is show that the limit (which, assuming the sequence converges, is unique) does, in fact, satisfy the given equation, not that it is the "only" solution.

Your recursion equation is $x_{n+1}= 2x_n/3+ \lambda/(3x_n^2)$. Take the limit of both sides as n goes to infinity and you have $x= 2x/3+ \lambda/3x^2$, where $x= \lim_{n\to\infty} x_n$. Multiply both sides of that by $3x^2$.

 

[Oshada]

Sorry, I meant how to show that the only possible value for limn→∞xn is cbrt(λ)! Also, if you could explain how to solve ii) of b) that would be great!

 

[Oshada]

Bump: Any ideas about b) are welcome.