1) I am stuck with the convergence of integral $$\int_0^{1/2} \frac{dr}{\log(r)}$$ I know that it equals $\mathrm{li}(1/2)$, but don't know how to prove that it converges. I have some trouble with understanding $\log(r)$ behavior around $0$.
I know that $\log(r)<r^{-a}$, for $a>0$ but it gives me nothing, because limits the bottom $1/\log(r)$.
2)$$\int_0^{1/2} \frac{r^{n-2}}{\log(r)}dr$$ where $n\geq2$. I think $r$ in the numerator only help our integral to converges.
3)$$\int_0^{1/2} \frac{r^{n(n-2)}}{\log^n(r)}dr$$ Well, I think if you give me some hints with the first problem I will get it all.
As $r \to 0+$ we have $\sqrt{r}/ \log r \to 0$.
Hence, for any $\epsilon > 0$, there exists $\delta > 0$ such that for $0 < r < \min(\delta,1/2) $ we have $|\sqrt{r}/\log r| < \epsilon$ and
$$\left|\frac{1}{\log r}\right| < \frac{ \epsilon}{\sqrt{r}}$$
Thus, $\displaystyle\int_0^{1/2}\frac{dr}{\log r}$ converges by the comparison test.