Let $g(t)=\lfloor t\rfloor$, and $f(t)=\frac1{t^\alpha+1}$ with $\alpha\neq1$ and $\alpha\in\Bbb R$. Find the value of the Riemann-Stieltjes integral $\int^n_1f(t)\,\mathrm dg(t)$ if it exists, where $n \in \Bbb Z$ is fixed.
I know, that integrating by parts gives $$ \int^{n}_{1} f dg = \int^{n}_{1} \frac{d \lfloor t \rfloor}{t^{\alpha}+1} = \frac{n}{n^{\alpha}+1}-\frac{1}{2}+ \alpha \int^{n}_{1} \frac{\lfloor t \rfloor t^{\alpha -1}}{(t^{\alpha }+1)^2}dt $$ I would like to know if the procedure is ok, although I am confused with the final result. I appreciate the help