Let's do some calculus! (55)

Let $S_n = \displaystyle \sum_{k=1}^n \dfrac{n}{n^2 + kn + k^2}$ and $T_n = \displaystyle \sum_{k=0}^{n-1} \dfrac{n}{n^2 + kn + k^2}$ for $n=1,2,3, \cdots$ .

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1) $S_n = T_n$ for some arbitrarily large $n$ .

2) $S_n < T_n$ for some arbitrarily large $n$ .

3) $S_n > T_n$ for some arbitrarily large $n$ .

4) $S_n < \dfrac{\pi}{3 \sqrt{3}}$ .

5) $S_n > \dfrac{\pi}{3 \sqrt{3}}$ .

6) $T_n < \dfrac{\pi}{3 \sqrt{3}}$ .

7) $T_n > \dfrac{\pi}{3 \sqrt{3}}$ .

8) $\displaystyle \lim_{n \to \infty} S_n$ converges but $\displaystyle \lim_{n \to \infty} T_n$ does not.

9) $\displaystyle \lim_{n \to \infty} T_n$ converges but $\displaystyle \lim_{n \to \infty} S_n$ does not.

10) Both $\displaystyle \lim_{n \to \infty} S_n$ and $\displaystyle \lim_{n \to \infty} T_n$ converge.

11) Neither of $\displaystyle \lim_{n \to \infty} S_n$ and $\displaystyle \lim_{n \to \infty} T_n$ converge.

Select the correct options from the choices given above and input your answer as the concatenation of the digits sequence-wise. For example, if the correct options are 1,4,3,2 input the number 1234 (One thousand two hundred and thirty-four). In case there is an option represented by a double-digit integer, simply put the digits in order following the same increasing sequence. For example, if the correct options are 4,3,12,11 input the number 341112 (Three hundred forty-one thousand one hundred and twelve or Three lakh forty-one thousand one hundred and twelve), i.e., (3)(4)(11)(12).

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