Let S_n=∑_ k=0 ^n (n)/(n^2+k n+k^2) and T_n=∑_ k=0 ^ n-1 (n)/(n^2+k...

Given, \begin{aligned} S_n & =\sum_{k=0}^n \frac{n}{n^2+k n+k^2} \\ & =\sum_{k=0}^n \frac{1}{n} \cdot\left(\frac{1}{1+\frac{k}{n}+\frac{k^2}{n^2}}\right) < \lim _{n \rightarrow \infty} \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} d x=\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{aligned} i.e., $\quad S_n < \frac{\pi}{3 \sqrt{3}}$ Similarly, $T_n>\frac{\pi}{3 \sqrt{3}}$.