For any positive integer n, let S_n : (0, $$\infty$$) $$\to$$ R be defined by $${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)} $$, where for any x $$\in$$ R, $${\cot ^{ - 1}}(x) \in (0,\pi )$$ and $${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$. Then which of the following statements is (are) TRUE?

Let $${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$ Then $${S_n}$$can take value(s)

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

For a positive integer $$n$$, let
$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$$. Then