Mathematics - Limits Question with Solution | TestHub

If $a_1$ is the greatest value of $f(x)$ where $f(x)=\frac{1}{2+[\sin x]}$ and $a_{n+1}=\frac{(-1{)}^{n+2}}{n+1}+a_n$ Then ${\lim }_{n\to \infty }{a}_n=$ $\_\_\_\_$