Exercise 6.1.12

Suppose we start with a point $x_0$, let’s expand around this point using Taylor expansion: $f(x)=f(x_0)+\frac{\mathit{df}}{\mathit{dx}}(x-x_0)+O({(x-x_0)}^2)$, we are looking for $x$ such that $f(x)=0$, so if we let $f(x)=0$, we have $x=x_0-\frac{f(x_0)}{\frac{\mathit{df}}{\mathit{dx}}}$, this point will be closer to the root than $x_0$, so we have an update formula here $x_{k+1}=x_k-\frac{f(x_k)}{\frac{\mathit{df}}{\mathit{dx}}}$

If we run Newton’s method on $f(x)=x^2+1$, we see the algorithm doesn’t converge, it oscillates from left to right of $x=0$.