Find the sum `cosec^(-1) sqrt10 + cosec^(-1) sqrt50 + cosec^(-1) sqrt(170) + .... + cosec^(-1) sqrt((n^(2) + 1) (n^(2) + 2n + 2))`

Explanation

Let `theta = cosec^(-1) sqrt((n^(2) + 1) (n^(2) + 2n + 2))`
or `cosec^(-2) theta =(n^(2) + 1) (n^(2) + 2n + 2)`
`=(n^(2) + 1)^(2) + 2n (n^(2) + 1) + n^(2) + 1`
`= (n^(2) + n + 1)^(2) + 1`
`rArr cot^(2) theta = (n^(2) + n + 1)^(2)`
`rArr tan theta = (1)/(n^(2) + n+ 1) = ((n+1) - n)/(1 + (n + 1) n)`
`rArr theta = tan^(-1) [((n+ 1) -n)/(1 + (n + 1)n)] = tan^(-1) (n + 1) - tan^(-1) n`
Thus, sum of `n` terms of the given series
`= (tan^(-1) 2 - tan^(-1)1) + (tan^(-1) 3 - tan^(-1)2)+ (tan^(-1) 4 - tan^(-1) 3) + ...+ (tan^(-1) (n + 1) - tan^(-1) n)`
`= tan^(-1) (n + 1) - pi//4`