$\int_0^1\frac{1}{r}\frac{1}{\left[\log\left(1+\frac{1}{r}\right)\right]^n}dr$ finite?

Question

Does someone have a hint for me why the integral $\int_0^1\frac{1}{r}\frac{1}{\left[\log\left(1+\frac{1}{r}\right)\right]^n}dr$ is finite? $n$ is a natural number greater than $1$.

Answer 1

The only singularity is at $r=0$ It is enough to consider the integral on an interval $(0,a)$ with $0<a<1$. Near $r=0$ we have $$ \log\Bigl(1+\frac1r\Bigr)=\log(1+r)-\log r\sim\log r, $$ and $$ \int_0^a\frac{dr}{r(\log r)^n}=\int_{-\infty}^{\log a}\frac{dt}{t^n}=\frac{1}{n-1}(\log a)^{1-n}. $$