View source for Mock AIME 3 Pre 2005 Problems/Problem 15

<math>15.</math> Let <math>\Omega</math> denote the value of the sum

<math>\sum_{k=1}^{40} \cos^{-1}\left(\frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}}\right)</math>

The value of <math>\tan\left(\Omega\right)</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>.