I know there are some post regarding this topic, this post is more to confirm if what I did is right and there's no mathematicall errors. I'm asked to compute de Laplace transform of the fuction floor of $x$, this is what I did:
Let $f(x)$ denote $\operatorname{floor}x$, and also let $L[f(x)]$ denote the Laplace transform of $f(x)$.
$L[f(x)]=\int_{0}^\infty e^{-px}f(x)dx=\sum_{n=0}^\infty n\int_{n}^{n+1} e^{-px}=\frac{1}{p}\sum_{n=1}^\infty ne^{-np} -ne^{-(n+1)p}=\frac{e^{-p}}{p}\sum_{n=0}^\infty (e^{-p})^n=\frac{e^{-p}}{p}(\frac{1}{1-e^{-p}})=\frac{1}{p(e^p-1)}$
Did my procedures were right?