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

1. 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...

2. Originally Posted by Krakowie
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...

What are l_i, n_i, L, M ???

3. I'm not sure if I'm right but I think these letter might just mean that there are even numbers in numerators and odd numbers in denominators of the fractions being summed. And that in the final result both numerator and demominator are odd.