Weighted Hardy Inequality in $l_2$

We study the behaviour of the smallest possible constant $d_n$ in weighted Hardy inequality $$\sum _{k=1}^n(\frac{1}{k}\sum _{j=1}^k{a}_j{)}^2{k}^{\epsilon }\le d(n,\epsilon )\sum _{k=1}^n{a}_k^2{k}^{\epsilon }$$ The exact rate of convergence of $d_n$ is established and the ``almost extremal’’ sequence is found.