7.2 Recursive sequences

E216

[0DK]Let \(f(x)=x-x^ 3\) and \(x_ 0∈{\mathbb {R}}\), and \((x_ n )_{n∈{\mathbb {N}}}\) a sequence defined by recurrence by \(x_{n+1}=f(x_ n)\). Prove that there is a \(𝜆{\gt}0\) such that if \(|x_ 0|{\lt}𝜆\) then \(x_ n→ 0\), while if \(|x_ 0|{\gt}𝜆\) then \(|x_ n|→ ∞\); and possibly calculate this \(𝜆\).

Hidden solution: [UNACCESSIBLE UUID ’0DM’]

E216

[0DN]Note:Babylonian method for square root. Let \(S{\gt}0\) and consider the sequence defined by recurrence as

\[ x_{n+1} = \frac 1 2 \left( x_ n + \frac{S}{x_ n} \right) \quad ; \]

show that \(x_ n→ \sqrt S\) and that, for \(S∈ [1/4,1]\) and \(x_ 0=1\), convergence is superquadratic, i.e.

\[ \left| x_ n-\sqrt{S} \right| ≤ 2^{1-2^ n }\quad . \]

Find a function \(f(x)\) (dependent on \(S\) ) such that the previous iteration can be seen as a Newton’s method, i.e.

\[ x - \frac{f(x)}{f'(x)} = \frac 1 2 \left( x + \frac{S}{x} \right). \]

Generalize the Babylonian method to find a root \(\sqrt[k]{S}\).

Hidden solution: [UNACCESSIBLE UUID ’0DP’]