I would greatly appreciate some help for this exercise in Kincaid and Cheney's Numerical Analysis: "apply Newton's method to $f(x)=x^2-r$ (where $r>0$). Prove that if $x_n$ has $k$ correct digits after the decimal point, then $x_{n+1}$ has at least $2k-1$ correct digits, as long as $r>0.006$ and $k\geq 1$".
In general, if $f''$ is continuous and $r$ is a simple root of $f$ it holds that $e_{n+1}=x_{n+1}-\sqrt{r}=\frac{1}{2}\frac{f''(\xi_n)}{f'(x_n)}e_n^2$, where $\xi_n$ is some number between $x_n$ and $\sqrt{r}$.
I suppose that the condition they're asking for is achieved (but I'm not sure) when
$e_{n+1} On the other hand, the textbook suggests to use $x_{n+1}=\frac{1}{2}(x_n+\frac{r}{x_n}).$ This expression can be transformed into one involving $e_{n+1}$ and $e_n$ but I don't know how to do it so that it would be more useful than the former bound. Thanks in advance for any help given.