Question

Let x_sub(1) = a>0 and x_sub(n+1) = x_sub(n) + 1/x_sub(n). Show that x_sub(n) diverges

Answer 1

Question 1. Let $x_{1} = a > 0$ and $x_{n + 1} = x_{n} + 1 / x_{n}$. Show that $x_{n}$ diverges.

Solution. Suppose there is $b = \lim x_{n} < \infty$. Note that $b > 0$, because $x_{1} > 0$ and $(x_{n})$ is increasing. Then consider the equality $x_{n + 1} = x_{n} + 1 / x_{n}$. As $n \to \infty$ the left-hand side of this equality tends to $b$, while the right-hand side tends to $b + 1 / b$. Since $b \neq b + 1 / b$, we obtain a contradiction.