2016/023) Show that $\lim_{n\to\infty} \frac{1}{n+1} + \frac{1}{n+2}\frac{1}{n+3} +\cdots + \frac{1}{6n} = ln 6$

we have $\lim_{n\to\infty} \frac{1}{n+1} + \frac{1}{n+2}\frac{1}{n+3} +\cdots + \frac{1}{6n}$
$= \lim_{n\to\infty} \sum_{r=1}^{5n} \frac{1}{n+r}$
$= \lim_{n\to\infty}\frac{1}{n} \sum_{r=1}^{5n} \frac{1}{1+\frac{r}{n}}$
$=\int_{x = 0}^{5}\frac{1}{1+x}dx =  \bigl[ln (1+x)\bigr]_1^5 = \ln 6 - \ln 1 = \ln 6 $