= Solution
Put $s=\sin x$, so $-1\le s\le1$. The printed summand is $(3s^n+s^{2n})/n$. If $|s|<1$, its absolute value is at most $3|s|^n+|s|^{2n}$, and the two <geometric series> converge. Thus there is <absolute convergence>.
For $s=1$ the <series> becomes $\sum4/n$, a divergent multiple of the <harmonic series>. For $s=-1$ it becomes
$$
3\sum_{n\ge1}\frac{(-1)^n}{n}+\sum_{n\ge1}\frac1n.
$$
The first <series> converges by the <alternating series test>, while the second diverges to $+\infty$, so their partial sums tend to $+\infty$. In particular, alternation of some terms does not imply convergence here. The complete classification is
$$
\boxed{\begin{array}{ll}x\notin\{\pi/2+k\pi:k\in\mathbb Z\}:&\text{absolute convergence},\\x\in\{\pi/2+k\pi:k\in\mathbb Z\}:&\text{divergence to }+\infty.\end{array}}
$$
\b[There are no conditionally convergent cases.]
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