Infinite series of $\lim_{n \to \infty}\sum_{r=n^2}^{(n+2)^2} \frac{1}{\sqrt{r}}$ Find Value of  $$S=\lim_{n \to \infty}\sum_{r=n^2}^{(n+2)^2} \frac{1}{\sqrt{r}}$$
$$S=\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots +\frac{1}{\sqrt{n^2+4n+4}}$$
But here i cannot express above sum in form of $$\lim_{n \to \infty} \frac{1}{n} \times \sum_{r=1}^{n}f(\frac{r}{n})$$

As you already wrote $\lim_{n \to \infty}\sum_{r=n^2}^{(n+2)^2} \frac{1}{\sqrt{r}}$,
$$S=\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots +\frac{1}{\sqrt{n^2+4n+4}}$$
When you take $n^2$ common out from the square root then your eq. becomes:
$$S=\frac{1}{n}\lim {n \to \infty}[1+ \frac{1}{\sqrt(1+ \frac{1}{n^2})}+......+\frac{1}{\sqrt(1+\frac{2}{n})^2}]$$
Can  you solve from here?
