I wanted to write a way of calculating a square root of a number in LaTeX. Unfortunately, as a beginner in this sphere, I do not know what math symbol means to repeat the process. To make it clear, here is my equation for finding a square root of a variable k
$$ \sqrt k = \frac{(n + \frac{k}{n})}{2} $$
After each calculation, a specific result is put instead of variable n, i.e: $$ \sqrt 5 = \frac{(5 + \frac{5}{5})}{2} = 3$$ $$ \sqrt 5 = \frac{(3 + \frac{5}{3})}{2} \approx 2.\overline{33} $$ $$ \sqrt 5 = \frac{(2.\overline{33} + \frac{5}{2.\overline{33}})}{2} \approx 2.23796 $$ $$ \sqrt 5 = \frac{(2.23796 + \frac{5}{2.23796})}{2} \approx 2.23606 $$ $$ \sqrt 5 = \frac{(2.23606797 + \frac{5}{2.23606797})}{2} \approx 2.23606797$$ $$ \sqrt 5 \approx 2.23606797... $$
FYI my English is not that good when explaining something, especially if it is Maths, so expect some weird or false vocabulary here and there...
Do not do that, instead define a sequence $(u_n)_n \in \mathbb{R}^\mathbb{N}$ with $$u_0 = x_0 = k$$ $$u_{n+1} = f(u_n) = f^n (u)$$ with $f(x) = \frac{x^2 + k}{2x}$.
Show that $u_n \rightarrow \sqrt{k}$ and it will be much cleaner.
For that show that $\sqrt{k}$ is a fixed point of $f$ and that your sequence is either decreasing or increasing wether you have $x_0 > \sqrt{k}$ or not