I've been studying Stieltjes-Integrals a bit, and came upon an integral where I don't see why the result given should be correct: The integral is $$\int_{1/2}^{n} \frac{d\lfloor t \rfloor}{t} = \frac{\lfloor n\rfloor}{n} + \int_{1/2}^{n} \frac{\lfloor t \rfloor}{t^2}dt.$$ From the theorems I recall, it holds that for $\alpha(x)$ with a continuous derivative $$\int_{a}^{b} f(x) d\alpha(x) = \int_{a}^{b} f(x)\alpha'(x)dx,$$ so shouldn't the integral above be negative, since $(\frac{1}{t})' = -\frac{1}{x^2}$?
Thanks for any help!
Integrating by parts gives $$ \int_{1/2}^n\frac{d\lfloor t\rfloor}{t}=\frac{\lfloor n\rfloor}{n}-\frac{\lfloor 1/2\rfloor}{1/2}-\int_{1/2}^n\lfloor t\rfloor d\left(\frac{1}{t}\right)=\frac{\lfloor n\rfloor}{n}+\int_{1/2}^n\frac{\lfloor t\rfloor}{t^2}dt $$