Question

Let
$S_n=\sum _{k=1}^n\frac{n}{n^2+kn+k^2}$ and $T_n=\sum _{k=0}^{n-1}\frac{n}{n^2+kn+k^2}$
for $n=1,2,3,..$ Then,

• A. $S_n<\frac{\pi }{3\sqrt{3}}$
• B. $S_n>\frac{\pi }{3\sqrt{3}}$
• C. $T_n<\frac{\pi }{3\sqrt{3}}$
• D. $T_n>\frac{\pi }{3\sqrt{3}}$

Answer 1

The correct option is C $T_n<\frac{\pi }{3\sqrt{3}}$

$S_n={\sum }_{k=1}^n\frac{n}{n^2+kn+k^2}$

$S_n=\frac{1}{n}{\sum }_{k=1}^n\frac{1}{1+\frac{k}{n}+{\left(\frac{k}{n}\right)}^2}$

$S_n={\int }_0^1\frac{1}{1+x+x^2}$

$S_n=\frac{\pi }{3\sqrt{3}}$