Is the series $\sum \frac{3 + \sin n}{n^2}$ convergent?

Question

How can I show if the following series converges?

$$\sum \frac{3 + \sin n}{n^2}$$

I can't use differential or integral calculus (hasn't been covered in my class yet.)

Answer 1

You know that $-1 \le \sin n \le 1$ for all $n$.

Use this to show that $\dfrac{3+\sin n}{n^2}$ is non-negative, and compare it with the series $\dfrac{4}{n^2}$.

Answer 2

Since $-1\le \sin n \le 1$, we have $\frac{2}{n^2} \le \frac{3+\sin n}{n^2} \le \frac{4}{n^2}$. Hence, try comparing your series to $\sum_{n = 0}^{\infty} \frac{2}{n^2}$ and $\sum_{n = 0}^{\infty} \frac{4}{n^2}$, both of which converge.