Mathematics

Let
$$\displaystyle { S }_{ n }=\sum _{ k=1 }^{ n }{ \dfrac { n }{ { n }^{ 2 }+kn+{ k }^{ 2 } }  }$$ and $$\displaystyle { T }_{ n }=\sum _{ k=0 }^{ n-1 }{ \dfrac { n }{ { n }^{ 2 }+kn+{ k }^{ 2 } }  }$$
for $$n=1,2,3,..$$ Then,


ANSWER

$${ T }_{ n }<\dfrac { \pi }{ 3\sqrt { 3 } }$$


SOLUTION
$$S_n=\sum_{k=1}^{n}\dfrac{n}{n^2+kn+k^2}$$

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

$$S_n=\int_{0}^{1}\dfrac{1}{1+x+x^2}$$

$$S_n=\dfrac{\pi}{3\sqrt{3}}$$
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Single Correct Medium Published on 17th 09, 2020
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