How can I solve the recursion tree T(n)=2T(n/2)+2/log(n)

Question

I would like to get some help I just don't know how to finish the sum :

I got:

$$T(n)=2^{k}T(\dfrac{n}{2^k}) + \sum_{i=1}^{log_{2}n} \dfrac{2^{i}}{log(\frac{n}{2^{i-1}})}$$

when $k=log_2n$ I get:

$$n+\sum_{i=1}^{log_2n} \dfrac{2^i}{log(\frac{n}{2^{i-1}})}$$