Sums, logarithms and harmonic numbers identity which I don't understand

I have the following equations, which is supposed to be correct, it was given by our teacher:

$\displaystyle 2^{\lfloor \lg n \rfloor- 1 } H_{n} - \frac{1}{2} = 2^{\lfloor \lg n \rfloor- 1 } \cdot (1 + (\frac{1}{2} +\frac{1}{3}) + (\frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}) + \cdot \cdot \cdot + (\frac{1}{2^{\lfloor \lg n \rfloor} } + \frac{1}{2^{\lfloor \lg n \rfloor + 1} } + \cdot \cdot \cdot + \frac{1}{n}) ) - \frac{1}{2} = 1 + \sum_{i} \frac{2l_{i}}{2n_{i} + 1} = \frac{2L + 1}{2M +1}$

But while I can see where the first part comes from perfectly well, I just don't know where did she get that summation from and the final result...