Find the value of: $\lim_{n \to \infty}\left(\frac{1}{n+1}+\cdots+\frac{1}{2n}\right)$

Question

Calculate $$\lim_{n \to \infty}\left(\dfrac{1}{n+1}+\cdots+\dfrac{1}{2n}\right).$$

I tried relating it to a Riemann sum, but I couldn't see how to do so.

Answer 1

For large $n$: \begin{align} \frac{1}{n+1} + \cdots + \frac{1}{2n} &\approx \int_n^{2n} \frac{1}{x} \,dx\\ &=\ln 2n - \ln n \\ &= \ln 2 \end{align}

Answer 2

$$\lim_{n\to\infty}\sum_{k=1}^n\frac1{n+k}=\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac1{1+\frac kn}=\int_1^2\frac{dx}x=\log2$$