Mathematics - Definite Integration Question with Solution | TestHub

Given, $$\begin{array}{rl}{S}_n& =\sum _{k=0}^n\frac{n}{n^2+kn+k^2}\\ & =\sum _{k=0}^n\frac{1}{n}\cdot \left(\frac{1}{1+\frac{k}{n}+\frac{k^2}{n^2}}\right)<\underset{n\to \infty }{\lim }\sum _{k=0}^n\frac{1}{n}\left(\frac{1}{1+\frac{k}{n}+{\left(\frac{k}{n}\right)}^2}\right)\\ & ={\int }_0^1\frac{1}{1+x+x^2}dx={\left[\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{2}{\sqrt{3}}\left(x+\frac{1}{2}\right)\right)\right]}_0^1\\ & =\frac{2}{\sqrt{3}}\cdot \left(\frac{\pi }{3}-\frac{\pi }{6}\right)=\frac{\pi }{3\sqrt{3}}\end{array}$$ i.e., $S_n<\frac{\pi }{3\sqrt{3}}$ Similarly, $T_n>\frac{\pi }{3\sqrt{3}}$.