Find the limit when $\displaystyle n\rightarrow\infty$ of the series
$\displaystyle \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+....+\fr ac{1}{2n}$
$\displaystyle \lim_{n \to \infty} \sum_{k=1}^{k=n} \frac1{k+n} = \lim_{n \to \infty} \frac1{n} \sum_{k=1}^{k=n} \dfrac1{\frac{k}{n}+1} = \int_0^{1} \frac1{x + 1} \, dx = \ln 2$
$\displaystyle \lim_{n \to \infty} \frac1{n} \sum_{k=1}^{k=n} \dfrac1{\frac{k}{n}+1} = \int_0^{1} \frac1{x + 1} \, dx$ is obtained by observing that the left hand side is a Riemann sum.