\displaystyle \lim_n \frac{a_n}{b_n} = \lim_n \left(\sqrt n / \sum_{k=1}^n \sqrt k \right) = \lim_n \left(\sum_{k=1}^n \sqrt \frac k n \right)^{-1} = \left(\sum_{k=1}^\infty \sqrt \frac k n \right)^{-1}