Series convergence or divergence?

Question

I'm having a hard time determining if the following series converges (absolutely?) or diverges:

$$\sum_{i=1}^n \frac 1 {2+\sin n}$$

I would really appreciate some help here. Thanks!

Answer 1

Note that we have $$0 \leq \dfrac{1}{2 + 1} \leq \frac{1}{2+\sin(k)}$$

for all $k \in \mathbb{N}$.

Since $\sum_{k \in \mathbb{N}} {\dfrac{1}{3}}$ diverges, by the Comparison test, the sum $$\sum_{k \in \mathbb{N}}\dfrac{1}{2+ \sin(k)}$$

diverges.

Answer 2

The terms do not approach $0$ since $1 \le 2+\sin n \le 3$, so $\dfrac 1 3 \le \dfrac 1 {2+\sin n}$. Therefore the series diverges.

Answer 3

Observe that $\lim_{n\rightarrow \infty} \frac1{2+\sin n}\neq0$ Hence the series diverges.

Answer 4

The "infinite" sum diverges ($\lim_{n \rightarrow \infty} c_n \not = 0$) . But your finite sum should converge, just add all the terms together up to n.