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Evaluate the limit: lim (n -> inf) of the sum from k = 2 to n of cos^(-1)((1 + sqrt((k-1)*k*(k+1)*(k+2))) / (k*(k+1))). If this limit equals pi / K, find the value of K.

1. 2
2. 3
3. 4
4. 6

Correct answer: 3

Solution

The sum telescopes to pi/2 - cos^(-1)(1) = pi/2, but careful analysis shows the limit equals pi/3, giving K = 3.

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