Determining whether $\sum_{n=2}^\infty\frac{\sin(n\pi/12)}{\ln n}$ converges

Question

I'm having issues with determining convergence/divergence of alternating series that use sine and cosine. I'm perfectly clear of how to handle ones with $(-1)^{n+1}$ (and similar) by performing the Absolute Convergence Test and by applying Leibnitz's theorem, but sine and cosine ones are a totally different story.

I simply don't know where to start on this one, for example.

$$\sum_{n=2}^\infty\frac{\sin\dfrac{n\pi}{12}}{\ln n}$$

Answer 1

We use the Dirichlet's test.

For that, we observe that $\frac{1}{\log (n)}$ is decreasing and tending to $0$ as $n\to \infty.$ Next we have to show that $$S_M=\sum_{n=1}^{M}\sin(n\pi/12)$$ is bounded.

For this use the following equation:

$$\sum_{k=1}^{n}\sin kx=\frac{\cos\left(\frac{1}2x\right)-\cos\left(n+\frac{1}2\right)x}{2\sin(x/2)},x\neq0,\pm\pi,\pm 2\pi,...$$