Question 12
Given a sequence of real numbers ($x_n$):
$$
x_1 = 1 \quad \text{and} \quad x_n = \frac{2n}{(n-1)^2} \sum_{i=1}^{n-1} x_i \quad \text{for all } n \ge 2.
$$
For each positive integer $n$, let $y_n = x_{n+1} - x_n$.
Show that the sequence ($y_n$) has finite limit as $n \to +\infty$.
$$
x_1 = 1 \quad \text{and} \quad x_n = \frac{2n}{(n-1)^2} \sum_{i=1}^{n-1} x_i \quad \text{for all } n \ge 2.
$$
For each positive integer $n$, let $y_n = x_{n+1} - x_n$.
Show that the sequence ($y_n$) has finite limit as $n \to +\infty$.