Find the value of `sum_(r=0)^ootan^(-1)(1/(1+r+r^2))`

Explanation

`tan^(-1) ((1)/(1 + r + r^(2))) = tan^(-1) ((r + 1 -r)/(1 + r(r + 1))) = tan^(-1) (r + 1) - tan^(-1) (r)`
`rArr underset(r =0) overset(n)sum [tan^(-1) (r + 1) - tan^(-1) (r)] = tan^(-1) (n + 1) - tan^(-1) (0) = tan^(-1) (n + 1)`
`rArr underset(r =0)overset(oo)sum tan^(-1) ((1)/(1 + r + r^(2))) = tan^(-1) (oo) = (pi)/(2)`