Question

$\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r}{n^2+nr}$ equals

Answer 1

$\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r}{n^2+nr}$

$=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^n\frac{\frac{r}{n}}{1+\frac{r}{n}}$

Using the definition of definite integral we get above expression

$=\underset{0}{\overset{1}{\int }}\frac{x}{1+x}dx$

$=\underset{0}{\overset{1}{\int }}(1-\frac{1}{1+x})dx$

$=1-\log 2$