MathNet 2015-part-07 — 185 problems (EN–VI) Question 119

Find the largest real number $k$ such that there exists a sequence of positive reals $\{a_{i}\}$ for which $\sum_{n=1}^{\infty} a_{n}$ converges but $\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n^{k}}$ does not.