TITLE: Limit and sequence involving tangent
QUESTION [1 upvotes]: Find the value of $$\lim_{n\to \infty}\sum_{r=2}^n\left(\cot\left(S_{r-1}\right) -\cot\left(S_r \right)\right)$$
  where  $$S_n=\sum_{r=1}^n \tan^{-1}\left( \frac{2\left(2r-1\right)}{4+r^2\left(r^2 -2r + 1\right)}\right)$$

My work 
I tried to convert into $$\tan^{-1}A - \tan^{-1}B$$
$$\tan^{-1}\left( \frac{\frac{\left(2r-1\right)}{2}}{1+\frac{r^2\left(r -1\right)^2}{4}}\right)$$
But it was unsuccessful I was not able to find it in this form .

REPLY [5 votes]: Let $a_r=\tan^{-1}\left(\dfrac {r^2}2\right)$.
$\displaystyle a_r-a_{r-1}=\tan^{-1}\left( \frac{2\left(2r-1\right)}{4+r^2\left(r^2 -2r + 1\right)}\right)$
Therefore, $S_n$, like the limit you are trying to find, are just a telescoping series.
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