This came up in a part of the proof.
$-\log(1-x)$ is $x$ and then want to calculate the error of this.
The idea is that taylor series of $-\log(1-x)=x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+...$ We have $|x|<1$.I know how to calculate Taylor expansion, however can't see the justification from saying it is x. Next it says what is the error of this.
Well, it has
$x\leq \int_{1-x}^{1} \dfrac{dt}{t} \leq \dfrac{1}{1-x} x$
However, can't understand how this is true.
This is due to trying to prove that $0 \leq \sum_{p\leq N} ((-log(1- \dfrac{1}{p})-\dfrac{1}{p}) \leq \sum_{p \leq N} \dfrac{1}{p(p-1)}$