TITLE: Sum of infinite terms of $\cot^{-1}(2)+\cot^{-1}\bigl(\frac{9}{2}\bigr)+\cot^{-1}(8)+\cot^{-1}\bigl(\frac{25}{2}\bigr)+\cot^{-1}(18)+...$
QUESTION [9 upvotes]: How can we find sum of infinite terms of the series: 
$$\cot^{-1}(2)+\cot^{-1}\biggl(\frac{9}{2}\biggr)+\cot^{-1}(8)+\cot^{-1}\biggl(\frac{25}{2}\biggr)+\cot^{-1}(18)+ \dots$$
The difference of argument is in A.P. but it does not seem to be valuable. Could someone provide some hint?

REPLY [6 votes]: Similarly as in Find sum of the Trignomertric series, you can use the identity
$$
\DeclareMathOperator{\arccot}{arccot}
\arccot(x) - \arccot(y) = \arccot\left(\frac{xy+1}{y-x}\right) 
$$
to conclude that
$$
 \arccot \frac{n+2}{n} - \arccot \frac{n}{n-2} = \arccot \frac{n^2}{2}
$$
which allows to write your series as a sum of two telescoping series.