$$\sum_{i=0}^{\log n} \frac{n}{\log\left(\frac{n}{2^i}\right)}$$
I'm having trouble seeing how this summation simplifies.
It seems it would be something like:
$$\frac{n}{\log(n)} + \frac{n}{\log\left(\frac{n}{2}\right)} + \frac{n}{\log\left(\frac{n}{4}\right)} + \frac{n}{\log\left(\frac{n}{8}\right)} + \cdots +\frac{n}{\log\left(\frac{n}{2^{\log(n)}}\right)}$$
Somehow this sum $\in O(n\log(\log n))$, and I'm not sure why. Can anyone help me understand?