I've got these two sequences
- $${x_n}{_+}{_1}={x_n}{_-}{_1}{x_n}^2$$ or
- $${x_n}{_+}{_1}={x_n}{x_n}{_-}{_1}^2$$
how can I know where do they converge to and what is the rate of convergence when they are not constant and also do converge (for sure)?
for the first one I guessed that it might converges to zero only and the second one to zero or one but I'm not really sure about that and I don't know how to show that.
we usually use $${\epsilon_n}={x_n}-c$$
$${\epsilon_n}{_+}{_1}+c=({\epsilon_n}{_-}{_1}+c)({\epsilon_n}+c)^2$$ so if it really converges to zero then c=0 and we get
$${\epsilon_n}{_+}{_1}=({\epsilon_n}{_-}{_1})({\epsilon_n})^2$$ and then I don't know how to show what the rate is.
we usually get something like that $${\epsilon_n}{_+}{_1}={\epsilon_n}{\epsilon_n}{_-}{_1}$$ and from that I can conclude what the order is. any help would be appreciated.
thanks.
Consider the logarithm, then the second equation as the easier example becomes $$ \ln|x_{n+1}|=\ln|x_n|+2\ln|x_{n-1}| $$ is a linear recursion. Its characteristic polynomial $q^2=q+2$ or $0=q^2-q-2=(q-2)(q+1)$ has roots $2$ and $-1$, so that $$ x_n=A^{2^n}\cdot B^{(-1)^n} $$ where the constants are determined from the initial values $x_0=AB$, $x_1=A^2B^{-1}$.
With $|A|<1$ one gets convergence to zero, $|A|=1$ and $B\ne1$ a bounded but not converging sequence and for $|A|>1$ a diverging sequence.