I'm in computer science, but unfortunately my mathematics skills leave a lot to be desired. Anyhow, I pulled up this sheet to help me, but I can't seem to figure out exactly how it works for steps (described in the document after the proof) 4-6.
In particular, could someone please explain how 2^n accounts for these aspects of the sum, and then, why we multiply by 1/2^(n+1)?
The paper I pulled up doesn't do a great job of explaining that.
(As a side note: Any suggestions on how to get better at this stuff? I really seem to be the village idiot....)
http://www2.isye.gatech.edu/~hsharp/math2420/harmonic.pdf
I sincerely appreciate any help!
They are using the fact that $\frac{1}{2^n+k} \geq \frac{1}{2^n+2^n} = \frac{1}{2^{n+1}}$ for any $k\leq 2^n$ to move from (4) to (5),(notice there are $2^n$ of such terms, that's why you see the $2^n \times \frac{1}{2^{n+1}})$
then they simply cancelled $2^n$ on both numerator and denominator to get from (5) to (6)