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.