How fast does the sequence converge? We have the sequence $$a_1=\frac{9}{4} \quad \qquad a_{n+1}=2\frac{a_n^3-a_n^2+1}{3a_n^2-4a_n+1}$$ that converges to $\frac{9}{4}$.
I want to check how fast the convergence is.
Do we have to calculate $\lim_{n\rightarrow \infty}\frac{|a_{n+1}-9/4|}{|a_n-9/4|}$ ?

You want to find how many terms it takes in order for $|a_n-\frac{9}{4}|<\epsilon$, where $\epsilon$ is some chosen positive constant. For example, how many terms would it take for $a_n-\frac{9}{4}<10^{-3}$?
