How can I compute this sum? I want to calculate the summation 
$$\sum_{i=0}^{\log_2 n - 1}\frac{1}{\log_2 n-i}$$
when $n$ is a power of $2$.
Even a reasonable estimate on lower bound and upper bound on this summation is fine for me. 
I know that we can establish the lower bound and upper bound using integration but I am not able to correctly establish what this bound will be.

Note for integer $m$ we have $$\sum_{k=0}^{m-1}\frac1{m-k}=\sum_{j=1}^m\frac1j=H_m\sim\ln m+\gamma,$$
where $H_m$ denotes the $m$th Harmonic number and $\gamma$ the Euler-Mascheroni constant.
