The value of `sum_(m=1)^ootan^(- 1)((2m)/(m^4+m^2+2))` is

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)`
D. None of these

Answer

A

Explanation

`sum_(m=1)^(n)tan^(-1)((2m)/(m^(4)+m^(2)+2))`
`=sum_(m=1)^(n)"tan"^(-1)(((m^(2)+m+1)-(m^(2)-m+1))/(1+(m^(2)+m+1)(m^(2)-m+1)))`
`=sum_(m=1)^(n)[tan^(-1)(m^(2)+m+1)-tan^(-1)(m^(2)-m+1)]`
`=tan^(-1)(n^(2)+n+1)-tan^(-1)1`
`=tan^(-1)[(n^(2)+n+1-1)/(1+(n^(2)+n+1)1)]`
`=tan^(-1)((n^(2)+n)/(2+n^(2)+n))`