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The sum of series sec^(-1) sqrt2 + sec^(...

The sum of series `sec^(-1) sqrt2 + sec^(-1).(sqrt10)/(3) + sec^(-1).(sqrt50)/(7) +... + sec^(-1) sqrt(((n^(2) + 1) (n^(2) -2n + 1))/((n^(2) -n + 1)^(2)))` is

Text Solution

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Let `S = sec^-1sqrt2+sec^-1(sqrt10/3)+sec^-1(sqrt50/7)+...+sec^-1sqrt(((n^2+1)(n^2-2n+2))/(n^2-n+1)^2)`
Here, `T_n = sec^-1sqrt(((n^2+1)(n^2-2n+2))/(n^2-n+1)^2)`
Let `sec^-1sqrt(((n^2+1)(n^2-2n+2))/(n^2-n+1)^2) = theta`
`=>sec theta =sqrt(((n^2+1)(n^2-2n+2))/(n^2-n+1)^2`
`=>sec^2theta = ((n^2+1)(n^2-2n+2))/(n^2-n+1)^2`
`=>sec^2theta = ((n^2+1)(n^2+1-2n+1))/(n^2-n+1)^2`
`=>1+tan^2theta = ((n^2+1)(n^2+1)-2n(n^2+1)+n^2+1)/(n^2-n+1)^2`
`=>1+tan^2theta = ((n^2+1-n)^2+1)/(n^2-n+1)^2`
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