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 $n=1,2,3,\dots$. 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

Answer: (A), (D)

The correct options are
A $S_n<\frac{\pi }{3\sqrt{3}}$
D $T_n>\frac{\pi }{3\sqrt{3}}$

$S_n<\underset{n\to \infty }{lim}{S}_n=\underset{n\to \infty }{lim}\sum _{k=1}^n\frac{1}{n}\cdot \frac{1}{1+k/n+(k/n{)}^2}$
$={\int }_0^1\frac{dx}{1+x+x^2}=\frac{\pi }{3\sqrt{3}}$
Now, $T_n>\frac{\pi }{3\sqrt{3}}$ as $h\sum _{k=0}^{n-1}f\left(kh\right)>{\int }_0^{1}f\left(x\right)dx>h\sum _{k=1}^{n}f\left(kh\right)$
Ans: A,D