The alternating series $\sum_{k=1}^{\infty} \frac{(-1)^k(2 – \sin k)}{2k}$ seems to be convergent, but Leibniz criterion does not apply

Question

I was looking for an example of convergent, alternating series $\sum_{k=1}^{\infty}(-1)^kb_k$ such that $\{b_k\}_{k=1}^{\infty}$ is not eventually monotone, so that Leibiniz criterion could not be applied. Preferably, one whose convergence is conditional (not absolute). So, I thought of $\sum_{k=1}^{\infty} \frac{(-1)^k(2 – \sin k)}{2k}$. WolframAlpha says that this series converges, and it is clearly not absolutely convergent. But I am trying to prove its convergence, and I haven't been successful so far. Does anyone have any ideas?

Answer 1

Note that

$$\sum_{k=1}^n\frac{(-1)^k \sin k}{2k} = \sum_{k=1}^n\frac{\cos (\pi k) \sin k}{2k} = \sum_{k=1}^n\frac{\sin ((\pi +1)k)}{2k}$$

and the right hand side series converges by Dirichlet's test.