Question

What kind of growth does this function have? \(f(n) = f(n-1) + \dfrac{1}{f(n-1)};~n\in\mathbb{N}\), where \(f(0) = 1\)

From http://www.reddit.com/r/CasualMath/comments/2jhwtm/is_fn_fn1_1fn1_f0_1_logarithmic_growth/

Answer 1

One answer said it seemingly approaches to \(\sqrt{2x}\), so \(O(\sqrt x)\)

But how can we show that?

I don't understand other answer, though.

Answer 2

Do you have any idea? @SithsAndGiggles