Limit with inverse trignometry

We have to find the value of
$$\lim_{n\to \infty}\sum_{k=2}^n\arccos\left(\frac{1+\sqrt{(k-1)k(k+1)(k+2)}}{k(k+1)}\right)$$

I tried it a lot but could not find any start.

Please help me in this .

Solutions Collecting From Web of "Limit with inverse trignometry"



Use Telescoping series

We have the identity $$\arccos \alpha -\arccos \beta = \arccos(\alpha \beta+\sqrt{1-\alpha^2}\sqrt{1-\beta^2})$$ Here we have that $\alpha =\frac{1}{k}$ and $\beta =\frac{1}{k+1}$. Then our limit transforms to $$\lim_{n\to \infty} \arccos(1/2)-\arccos(1/3) +\arccos(1/3)-\arccos(1/4)+\cdots$$ This limit can then be easily calculated. Hope it helps.