Question 2
Let the real sequence $(x_n)$ be defined by
$$x_1=\frac12,\qquad x_n=\frac{\sqrt{x_{n-1}^2+4x_{n-1}}+x_{n-1}}2\quad(n\ge2).$$
For each positive integer $n$, put $y_n=\sum_{i=1}^n1/x_i^2$. Prove that $(y_n)$ has a finite limit as $n\to\infty$ and find that limit.
$$x_1=\frac12,\qquad x_n=\frac{\sqrt{x_{n-1}^2+4x_{n-1}}+x_{n-1}}2\quad(n\ge2).$$
For each positive integer $n$, put $y_n=\sum_{i=1}^n1/x_i^2$. Prove that $(y_n)$ has a finite limit as $n\to\infty$ and find that limit.