∑_m=1ⁿ tan^-1 [2m] / [m⁴ + m² + 2] =

1) tan^-1 [2m] / [n² + n + 2]

2) tan^-1 [n² + n] / [n² + n + 2]

3) tan^-1 [n² + n + 2] / [n² + n]

4) None of these

Solution: (2) tan^-1 [n² + n] / [n² + n + 2]

∑_m=1ⁿ tan^-1 [2m] / [m⁴ + m² + 2]

= ∑_m=1ⁿ tan^-1 [2m] / [1 + (m² + m + 1) (m² – m + 1)]

= ∑_m=1ⁿ tan^-1 [(m² + m + 1) – (m² – m + 1)] / [1 + (m² + m + 1) (m² – m + 1)]

= ∑_m=1ⁿ tan^-1 [(m² + m + 1) – tan^-1 (m² – m + 1)]

= tan^-1 3 – tan^-1 1 + tan^-1 7 – tan^-1 3 + tan^-1 13 – tan^-1 7 + ….. [tan^-1 (n² + n + 1) – tan^-1 (n² – n + 1)]

= tan^-1 (n² + n + 1) – tan^-1 1

= tan^-1 [n² + n] / [2 + n² + n]