Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ and the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_{n+1} := x_n - \tau f'(x_n)$. We suppose in this question that $0 < x_0 < 1/\tau$. a) Justify that the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ is decreasing, with strictly positive values, and satisfies $x_{n+1} = x_n(1 - \tau x_n)$ for all $n \in \mathbb{N}$. b) Justify that $x_n \rightarrow 0$ when $n \rightarrow \infty$. c) Show that $1/x_{n+1} = 1/x_n + \tau/(1 - \tau x_n)$ for all $n \in \mathbb{N}$. Deduce that $x_n \leq x_0/(1 + n\tau x_0)$.
Consider the function
$$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$
and the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_{n+1} := x_n - \tau f'(x_n)$.\\
We suppose in this question that $0 < x_0 < 1/\tau$.\\
a) Justify that the sequence $\left(x_n\right)_{n \in \mathbb{N}}$ is decreasing, with strictly positive values, and satisfies $x_{n+1} = x_n(1 - \tau x_n)$ for all $n \in \mathbb{N}$.\\
b) Justify that $x_n \rightarrow 0$ when $n \rightarrow \infty$.\\
c) Show that $1/x_{n+1} = 1/x_n + \tau/(1 - \tau x_n)$ for all $n \in \mathbb{N}$. Deduce that $x_n \leq x_0/(1 + n\tau x_0)$.