# To find approximate sqrta we can use Newton's method to approximately solve the equation x^2−a=0 for x, starting from some rational x_0. Newton's method in general is only locally convergent, so we have to be careful with initialization.Show that in this case, the method always converges to something if x_0≠0.

To find approximate $\sqrt{a}$ we can use Newton's method to approximately solve the equation ${x}^{2}-a=0$ for $x$, starting from some rational ${x}_{0}$.
Newton's method in general is only locally convergent, so we have to be careful with initialization.
Show that in this case, the method always converges to something if ${x}_{0}\ne 0$.
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Dillon Levy
The most convenient method to analyze this iteration is to explore the fraction
${\theta }_{n}=\frac{{x}_{n}-\sqrt{a}}{{x}_{n}+\sqrt{a}}$
as you will then find that