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

(a) Show that … in Example 2 is the same as

(a) Show that $x_{n + 1} = 2 x_{n} - 2 x_{n}^{2}$ in Example 2 is the same as $(1 - 2 x_{n + 1}) = (1 - 2 x_{n})^{2}$ .
(b) Prove divergence if $∣1 - 2 x_{0} ∣ > 1$ . Prove convergence if $∣1 - 2 x_{0} ∣ < 1$ or $0 < x_{0} < 1$ .

This exercise builds on another exercise in its source book, which isn't shown here.

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

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