## University Calculus: Early Transcendentals (3rd Edition)

Converges to $\dfrac{-\pi}{6}$
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 $\dfrac{-\pi}{6}$