Find the limit

**varunnayudu** · December 19th 2008, 03:51 AM

Find the limit when $n\rightarrow\infty$ of the series
$\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+....+\fr ac{1}{2n}$

**Isomorphism** · December 19th 2008, 04:00 AM

Originally Posted by varunnayudu

Find the limit when $n\rightarrow\infty$ of the series
$\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+....+\fr ac{1}{2n}$

$\lim_{n \to \infty} \sum_{k=1}^{k=n} \frac1{k+n} = \lim_{n \to \infty} \frac1{n} \sum_{k=1}^{k=n} \dfrac1{\frac{k}{n}+1} = \int_0^{1} \frac1{x + 1} \, dx = \ln 2$

$\lim_{n \to \infty} \frac1{n} \sum_{k=1}^{k=n} \dfrac1{\frac{k}{n}+1} = \int_0^{1} \frac1{x + 1} \, dx$ is obtained by observing that the left hand side is a Riemann sum.

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