Let us say we have the sequence:
$$\epsilon_{n+1}=\epsilon_n-\frac{1+\epsilon_n-e^{2\epsilon_n}}{1-2e^{2\epsilon_n}}$$
In order to find the rate of convergence, I have to do taylor series here.
Problem is, How do I know when do I stop?
I tried to take the first order of $e^{2\epsilon_n}$ and at the end I received:
$$\epsilon _{n+1}=\frac{4\epsilon_n\left(\epsilon_n-1\right)}{4\epsilon_n-1}$$
Which means my Rate of convergence should be 1 with $|A|=k$, for some k constant.
Although by the answer, the rate of convergence is 2.
To fix this, if I took the taylor series up to the second term, which means:
$$e^{2\epsilon_n}=1+2\epsilon_n+\frac{\left(2\epsilon_n\right)^2}{2!}$$
Then I will get the good answer, which rate of convergence of 2.
What is the reason of this? when do I know where to stop at taylor series so my rate of convergence rate will be accurate?