Suppose we want to calculate the square root of some positive real number $k>0$. Using the Newton-Raphson method, we know that the recursive sequence $\{a_n\}_{n=1}^{\infty}\subseteq\mathbb{R}$ that satisfies: $$a_1=1\ \ \ ,\ \ \ a_{n+1}=\frac{a_n^2+k}{2a_n}$$ will converge to $\sqrt{k}$ as desired.
My question is, given some $\varepsilon>0$, can we find an $m\in\mathbb{N}$ that for sure satisfies the inequality: $|a_m-\sqrt{k}|<\varepsilon$ ? (without knowing the value of $\sqrt{k}$ that we seek).
I know that in the general case there is no such nice estimation for the Newton-Raphson method, though I wonder, maybe there is one for that very simple case?
Thanks!