Solve the recurrence: $T(n)=2T(\frac{n}{2})+\frac{n}{lgn}.$

$[$Hint: ${\displaystyle \underset{i=1}{\overset{m}{}}}\frac{1}{i}=lnm+(1).$

First, use the changing variables method $($Heuristic $2)$ to transform the recurrence into a new form.

Use the Expansion method on the new form to derive an intelligent guess of $T(n).$ Then, verify the

correctness of the intelligent guess with mathematical induction. $]$