Question 1
Let $(x_n)$ be a real sequence with $x_1\in(0,1/2)$ and
$$x_{n+1}=3x_n^2-2n x_n^3\qquad(n\ge1).$$
a) Prove that $\lim_{n\to\infty}x_n=0$.
b) For $n\ge1$, put $y_n=x_1+2x_2+\cdots+nx_n$. Prove that $(y_n)$ has a finite limit.
$$x_{n+1}=3x_n^2-2n x_n^3\qquad(n\ge1).$$
a) Prove that $\lim_{n\to\infty}x_n=0$.
b) For $n\ge1$, put $y_n=x_1+2x_2+\cdots+nx_n$. Prove that $(y_n)$ has a finite limit.