Evaluate $\displaystyle \sum_{n=0}^{\infty}\cot^{-1}(n^2+n+1)$
and this one is similar.
Evaluate $\displaystyle \sum_{n=0}^{\infty}\tan^{-1}\left(\frac{1}{n^2+n+1}\right)$
I presume that these are not problems that you do not know the answer to.
If this is the case please put them in the "Maths Games and Puzzles" forum
where we will talk about them at our leisure rather than reduce the time
we spend helping people with problems that they can't do.
Thank you
RonL