Does $\frac{1}{n+2+\sin(n)}$ converge to $0$?

Question

Does $1/(n+2+\sin(n))$ converge to 0?

I know that I will have to find the sum from $n=0$ to $n=\infty$, but how do i show whether this function/sequence converges to $0$?

Answer 1

$$\sum_{n=1}^{\infty}\frac{1}{n+2+\sin(n)}\geq \sum_{n=1}^{\infty}\frac{1}{n+3}$$

Answer 2

$n+2+sin(n)$ converges to infinity (because $n$ approaches infinity and $2+sin(n)$ is bounded between $[1,3]$), so $\frac{1}{n+2+sin(n)}$ approaches $0$

Answer 3

$$ \frac{1}{n+3} \le \frac{1}{n+2+\sin n} \le \frac{1}{n+1} $$ and both sides go to $0$. Also $$ \sum_{n\ge 0}\frac{1}{n+2+\sin n} \ge \sum_{n\ge 0} \frac{1}{n+3} \to \infty. $$