InterviewSolution
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InterviewSolution
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`sum_(m=1)^(n) tan^(-1) ((2m)/(m^(4) + m^(2) + 2))` is equal toA. `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 |
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Answer» Correct Answer - A 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))` |
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