Convergence of the series with sum of sines in denominator Does this series converge?
$$
\sum_{n=1}^{\infty} \frac{1}{\sin 1 + \sin 2 + \ldots + \sin n}
$$
What test do I have to use for it?

We have:
$$ \sin(1)+\sin(2)+\ldots+\sin(n) = \text{Im}\sum_{j=1}^{n}e^{ji} = \frac{\sin\left(\frac{n}{2}\right)\sin\left(\frac{n+1}{2}\right)}{\sin\left(\frac{1}{2}\right)}\tag{1}$$
hence:
$$ \sum_{n=1}^{N}\frac{1}{\sin(1)+\sin(2)+\ldots+\sin(n)}=2\sin\left(\frac{1}{2}\right)\sum_{n=1}^{N}\frac{1}{\cos\left(\frac{1}{2}\right)-\cos\left(n+\frac{1}{2}\right)}\tag{2}$$
but since the sequence given by $a_n = \exp\left(\frac{i}{2}+in\right)$ is dense in the unit circle, the general term of our series is not even bounded, hence the series cannot be convergent. An alternative proof, following Kelenner's comment below, comes from noticing that $\left|\frac{1}{\sin(1)+\sin(2)+\ldots+\sin(n)}\right|\geq\sin\left(\frac{1}{2}\right).$
