`sum_(m=1)^(n) tan^(-1) ((2m)/(m^(4) + m^(2) + 2))` is equal to

Options

A. `tan^(-1) ((n^(2) + n)/(n^(2) + n + 2))`
B. `tan^(-1) ((n^(2) -n)/(n^(2) - n + 2))`
C. `tan^(-1) ((n^(2) + n + 2)/(n^(2) + n))`
D. none of these

Answer

A

Explanation

We have `underset(m=1)overset(n)sum tan^(-1) ((2m)/(m^(4) + m^(2) + 2))`
`= underset(m =1)overset(n)sum tan^(-1) ((2m)/(1+(m^(2) + m +1)(m^(2) -m +1)))`
`= underset(m=1)overset(n)sum tan^(-1) (((m^(2) + m + 1) -(m^(2) - m + 1))/(1 + (m^(2) + m + 1) (m^(2) - m + 1)))`
`= underset(m=1)overset(n)sum [tan^(-1) (m^(2) + m+1) - tan^(-1) (m^(2) - m + 1)]`
`= (tan^(-1) 3 - tan^(-1)1) + (tan^(-1) 7 - tan^(-1) 3) + (tan^(-1) 13 - tan^(-1) 7) +....+ [tan^(-1) (n^(2) + n + 1) -tan^(-1) (n^(2) -n + 1)]`
`= tan^(-1) (n^(2) + n + 1) - tan^(-1) 1`
`= tan^(-1) ((n^(2) + n)/(2 + n^(2) + n))`