Find the value of .

MATHEMATICS · Subjective · AoPS · Mathematics · Sequences & Series · Sequences & Series

Problem

Find the value of lim⁡n→∞∑k=2ncos⁡−1(1+(k−1)k(k+1)(k+2)k(k+1))\lim_{n\to\infty}\sum_{k=2}^{n}\cos^{-1}\left(\frac{1+\sqrt{(k-1)k(k+1)(k+2)}}{k(k+1)}\right).

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