Find the value
`underset(n rarr oo)("lim") underset(k =2)overset(n)sum cos^(-1) ((1 + sqrt((k -1) k(k + 1) (k + 2)))/(k(k + 1)))`

Explanation

`cos^-1((1+sqrt(k(k-1)(k+1)(k+2)))/(k(k+1)))`

`=cos^-1((1+sqrt(k(k^2-1)(k+2)))/(k(k+1)))`

`=cos^-1(1/(k(k+1))+sqrt(1-1/k^2)sqrt(1-1/(k+1)^2))`

`=cos^-1 (1/(k+1)) - cos^-1(1/k)`

`:. sum_(k=2)^n cos^-1 (1/(k+1) - 1/k) = cos^-1(1/3) - cos^-1(1/2) +cos^-1(1/4) - cos^-1(1/3)+cos^-1(1/5) - cos^-1(1/4)+...+cos^-1 (1/(n+1)) - cos^-1(1/n)`

`=cos^-1 (1/(n+1)) - cos^-1(1/2)`

`:. Lim_(n->oo) sum_(k=2)^n cos^-1 (1/(k+1) - 1/k) = Lim_(n->oo) cos^-1 (1/(n+1)) - cos^-1(1/2) `

`=cos^-1(0) - pi/3`

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