Question 60
Consider the real sequence $(x_n)$ such that $x_1 \in (0, \frac{1}{2})$ and
$$
x_{n+1} = 3x_n^2 - 2n x_n^3, \quad \forall n \ge 1.
$$
Part (a). Prove that $\lim_{n \to \infty} x_n = 0$.
Part (b). For each $n \ge 1$, let $y_n = x_1 + 2x_2 + \dots + n x_n$. Prove that $(y_n)$ converges.
$$
x_{n+1} = 3x_n^2 - 2n x_n^3, \quad \forall n \ge 1.
$$
Part (a). Prove that $\lim_{n \to \infty} x_n = 0$.
Part (b). For each $n \ge 1$, let $y_n = x_1 + 2x_2 + \dots + n x_n$. Prove that $(y_n)$ converges.