Answer
Converges to $\dfrac{-\pi}{6}$
Work Step by Step
Since, we have $f(x)=\cos^{-1} x$
Here, $ s_n=[f(\dfrac{1}{2}-f(\dfrac{1}{3}]+[f(\dfrac{1}{3}-f(\dfrac{1}{4}]......[f(\dfrac{1}{k+1}-f(\dfrac{1}{k+2}]$
and $s_n=\cos^{-1}{\dfrac{1}{2}}-\cos ^{-1} \dfrac{1}{k+2}$
Thus, we have $\lim\limits_{n \to \infty} s_n=\lim\limits_{n \to \infty} [\cos^{-1}{\dfrac{1}{2}}-\cos ^{-1} \dfrac{1}{k+2}]=[\dfrac{\pi}{3}-\dfrac{\pi}{2}]=\dfrac{-\pi}{6}$
Thus, the given series converges to $\dfrac{-\pi}{6}$
