Another Summation Series

$S = \displaystyle\sum_{n=1}^∞ cot^{-1} ( n^{2} + \frac{3}{4})$

$S = \displaystyle\sum_{n=1}^∞ cot^{-1} ( \dfrac{4n^{2} + 3}{4})$

$S = \displaystyle\sum_{n=1}^∞ tan^{-1} (\dfrac{4}{4n^{2} + 3})$

$S = \displaystyle\sum_{n=1}^∞ tan^{-1} (\dfrac{1}{n^{2} + \frac{3}{4}})$

$S = \displaystyle\sum_{n=1}^∞ tan^{-1} (\dfrac{1}{1 + n^{2} - \frac{1}{4}})$

$S = \displaystyle\sum_{n=1}^∞ tan^{-1} (\dfrac{(n + \frac{1}{2})- (n - \frac{1}{2})}{1 + (n+\frac{1}{2})(n - \frac{1}{2})})$

$S = \displaystyle\sum_{n=1}^∞ tan^{-1}(n + \frac{1}{2}) - tan^{-1}(n - \frac{1}{2})$

$S = \displaystyle\lim_{n\rightarrow ∞} tan^{-1}(n + \frac{1}{2}) - tan^{-1}\frac{1}{2}$

$S = tan^{-1} 2$

Therefore,

$[1000S] = 1107$