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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!

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Integrating by parts gives $$ \int_{1/2}^n\frac{d\lfloor t\rfloor}{t}=\frac{\lfloor n\rfloor}{n}-\frac{\lfloor 1/2\rfloor}{1/2}-\int_{1/2}^n\lfloor t\rfloor d\left(\frac{1}{t}\right)=\frac{\lfloor n\rfloor}{n}+\int_{1/2}^n\frac{\lfloor t\rfloor}{t^2}dt $$

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  • $\begingroup$ Thanks! I hab the formula for integration by parts wrong in mind, but of course it makes sense like that! $\endgroup$
    – Gecko1111
    Jun 20 '20 at 13:50

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