= Solution
Let $r=|\sin x|\in[0,1]$. For $r<1$, the absolute summand $r^n/\sqrt n$ is at most $r^n$, so a <geometric series> comparison proves <absolute convergence>. For $r=1$, the <series> is $\sum(-1)^n/\sqrt n$: its positive term sizes decrease to zero, so the <alternating series test> proves convergence. Its absolute <series> is a divergent <p-series> with exponent $1/2$. Hence
$$
\boxed{\begin{array}{ll}x\notin\{\pi/2+k\pi:k\in\mathbb Z\}:&\text{absolute convergence},\\x\in\{\pi/2+k\pi:k\in\mathbb Z\}:&\text{conditional convergence}.\end{array}}
$$
\b[It converges for every real $x$ and never diverges.]
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