Let $1<c\le 2$. I have come across an equation that uses the fact that $O(\sum_{n=1}^\infty \frac{\log n}{n^c}) = O(\frac{1}{(c-1)^2}).$
I think the idea is to use that $O(\int_1^\infty \frac{\log w}{w^c})= O(\frac{1}{(c-1)^2})$, which I can show by splitting the integral into intervals less than or greter than $e^{\frac{1}{c-1}}$ and using that $1-c<0$.
My question is how do we know that the infinites series here and the infinite integral are of the same order?