Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.7 Newton’s Method (and Chaos), #27 · Calculus · Applications of derivatives

If …, each new … is …y_{n+1}=x_{n+1}^2+1=\frac{1}{4}\left(x_n-\frac{1}{x_n}\right)^2+1.… Show that this equals

This problem is about $x^{2} + 1 = 0$ and chaos.

If $y = x^{2} + 1$ , each new $y$ is
$y_{n + 1} = x_{n + 1}^{2} + 1 = \frac{1}{4} (x_{n} - \frac{1}{x _{n}})^{2} + 1.$
Show that this equals $y_{n}^{2} /4 (y_{n} - 1)$ .

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.7 Newton’s Method (and Chaos), #27

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