Converting a Summation to an integral

Question

Please how do I convert this summation $$ \frac{r-1}{n} \sum_{i=r}^n \frac{1}{i-1} $$ to the integral $$ x \int_x^1 \frac{1}{t} dt = -x \ln x? $$ by substituting $x = r/n$, $t=1/n$ and $dt =1/n$. This integral is from the Secretary Problem.

Answer 1

For a fixed $r$, we have

$$ \frac{r-1}{n} \sum_{i=r}^n \frac{1}{i-1} =( r-1 )\sum_{i=r}^n \left(\frac{1}{i-1}\frac{1}{n}\right) $$ When $n$ goes to $+\infty$, we have

$$ (r-1) \sum_{i=r}^n \left(\frac{1}{i-1}\frac{1}{n}\right) \to (r-1)\int_{r-1}^1 \frac{1}{t}dt $$