Find a sequnce $\{a_n\}_{n=1}^{\infty}\in l_2$ such that $\sum_{n=1}^{\infty} a_n/\sqrt{n} = \infty$. As stated in title, I am curious how to construct a sequnce $\{a_n\}_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty} a_n^2 < +\infty$ but $\sum_{n=1}^{\infty}\frac{a_n}{\sqrt{n}} = \infty$.
This problem arises when I try to show that $f(x) = \sum_{i=1}^{\infty} \frac{x_n}{\sqrt{n}}$ defined on $l_2$ space is not bounded.

Try $a_n = (\sqrt{n}\log(n))^{-1},$ for $n\geq 2$.
