improper integral for a function involving floor function I am trying to study the improper integrability of $\int_{1}^{\infty}1-\frac{\lfloor x\rfloor}{x} d{x}$. I tried the definition of the improper integral as a limit with no success. Any hint? 

Here is a hint:


*

*Write (with $\{x\}$ being the fractional part of $x$):
$$\int_1^n \frac{\{x\}}{x}dx = 
\sum_{k=1}^{n-1} \int_k^{k+1} \frac{\{x\}}{x}dx = 
\sum_{k=1}^{n-1} \int_0^1 \frac{\{x+k\}}{x+k}dx$$

*Check that $\{x+k\} = x$ for $x \in (0,1)$

*Get an explicit formula for the last $\sum\int$, you should almost be there
