Find the value
`lim_(n rarr oo) sum_(k=2)^(n) cos^(-1) ((1 + sqrt((k -1) k(k + 1) (k + 2)))/(k(k + 1)))`

Explanation

`S_(n -1) = underset(k=2)overset(n)sum cos^(-1)((1 + sqrt((k-1) k(k + 1) (k + 2)))/(k(k+ 1)))`
`= cos^(-1). ((1)/(k).(1)/((k+1)) + (sqrt((k-1) k(k+1) (k+2)))/(k(k+1)))`
`= cos^(-1) ((1)/(k).(1)/((k+1)) + sqrt(1-(1)/(k^(2))) sqrt(1-(1)/((k+1)^(2))))`
`= cos^(-1).(1)/(k+1) - cos^(-1).(1)/(k) = t_(k)`
`( :' (1)/(k+ 1) lt (1)/(k), " so "cos^(-1). (1)/(k + 1) gt cos^(-1).(1)/(k))`
Substituting `k = 2, 3, 4`..., we get
`t_(2) = cos^(-1) ((1)/(3)) - cos^(-1) ((1)/(2))`
`t_(3) = cos^(-1) ((1)/(4)) - cos^(-1) ((1)/(3))`
`t_(n) = cos^(-1) ((1)/(n + 1)) - cos^(-1) ((1)/(n))`
`:. S_(n -1) = cos^(-1) ((1)/(n + 1)) - cos^(-1) ((1)/(2))`
`:. S = underset(n rarr oo)("lim") [cos^(-1) ((1)/(n + 1)) - cos^(-1) ((1)/(2))]`
`= cos^(-1) (0) - cos^(-1) ((1)/(2))`
`= (pi)/(2) - (pi)/(3) = (pi)/(6)`