The recurrent sequence $u_{n+1}=u_n+a_n/u_n$ is convergent if and only if $\sum a_n$ is finite

Question

Let $u_0 > 0$ be a real number and let $(a_n)$ be a sequence of strictly positive real numbers. Define the sequence $(u_n)$ by : $u_{n+1} = u_n + (a_n/u_n)$ Show that $(u_n)$ is convergent if and only if $\sum a_n < \infty$

I don't know how to approach.

Answer 1

Some hints:

Note that the $u_n$ are non decreasing, so $u_n \ge u_0$ for all $n$.

Note that $\sum_{k=0}^n a_k = \sum_{k=0}^n u_k(u_{k+1}-u_k)$.

Suppose $a_k$ are summable. Then $\sum_{k=0}^n a_k \ge u_0 \sum_{k=0}^n (u_{k+1}-u_k) = u_0 (u_{n+1}-u_0)$.

Now suppose $u_n$ converges, that is $u_n \to u$. Note that the $u_n$ are non decreasing, hence $u_n \le u$ for all $n$. Then $\sum_{k=0}^n a_k \le u \sum_{k=0}^n (u_{k+1}-u_k) = u (u_{n+1}-u_0)$.