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

Options

A. `pi/4`
B. `pi/8`
C. `pi/10`
D. `pi/12`

Answer

A

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)`