Evaluate $\lim_{n\to \infty}\left(\frac{1}{\sqrt{n(n+1)}}+\frac{1}{\sqrt{(n+1)(n+2)}}+\cdots+\frac{1}{\sqrt{(2n-1)2n}}\right)$

Hint: $$\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}<\frac{1}{\sqrt{n(n+1)}}+\frac{1}{\sqrt{(n+1)(n+2)}}+\cdots+\frac{1}{\sqrt{(2n-1)(2n)}}$$

$$<\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}$$ and use $\lim_{n \rightarrow \infty} (\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n})=\ln 2$

HINT

The hint provided by Chinnapparaj R is the key point, to evaluate the bounding sum we can use that

$$\frac1n+\frac1{n+1}+\ldots+\frac1{2n}=\sum_{k=1}^n \frac{1}{n+k}=\frac1n \sum_{k=1}^n \frac{1}{1+\frac kn}$$

which is a Riemann's sum.