Difficulties proving a sequence converges to infinity

Question

I was given an exercise to show that the sequence defined by: $$a_{n+1} = {a_n} + \frac{1}{2{a_n}}, {a_0} > 0$$

prove that: $$\lim_{n \to \infty} {a_n} = +\infty$$

I have proven by induction that the sequence is constantly increasing, and therefore it has a limit which is or a certain number, or +/- infinity.

Using the ratio test, it looks like the

$$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{{a_n} + \frac{1}{2{a_n}}}{a_n} = \lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{1}{2{a_n}^2} = 1 + 0 = 1$$

And therefore the ratio test inconclusive.

I've got 2 questions:

1.Am I using the ratio test correctly?

2.If I'm using the ratio test correctly, and it is really inconclusive, any hint how to continue?

Thank you.

Answer 1

We have: $$ a_{n+1}^2 = a_n^2 + 1 + \frac{1}{4 a_n^2}\geq a_n^2 + 1 $$ hence induction gives $a_n^2\geq n$ and $a_n\geq \sqrt{n}$ is enough to ensure $a_n\to +\infty$.

Answer 2

Hint: Your sequence is increasing. If it is bounded above then it has a finite limit which is a strictly positive number. What can you say about this presumed limit? Otherwise, the sequence is unbounded above.