Question 169
A sequence of real numbers $(x_n)$ is given by
$$
x_1 = \frac{1}{2} \quad \text{and} \quad x_n = \frac{\sqrt{x_{n-1}^2 + 4x_{n-1}} + x_{n-1}}{2} \quad \text{for all } n \ge 2.
$$
For each non-negative integer $n$, let
$$
y_n = \sum_{i=1}^{n} \frac{1}{x_i^2}.
$$
Show that the sequence $(y_n)$ has a finite limit when $n \to \infty$. Find this limit.
$$
x_1 = \frac{1}{2} \quad \text{and} \quad x_n = \frac{\sqrt{x_{n-1}^2 + 4x_{n-1}} + x_{n-1}}{2} \quad \text{for all } n \ge 2.
$$
For each non-negative integer $n$, let
$$
y_n = \sum_{i=1}^{n} \frac{1}{x_i^2}.
$$
Show that the sequence $(y_n)$ has a finite limit when $n \to \infty$. Find this limit.