= Solution
The summands repeat the six-term pattern
$$
\frac{\sqrt3}{2},\quad\frac{\sqrt3}{2},\quad0,\quad-\frac{\sqrt3}{2},\quad-\frac{\sqrt3}{2},\quad0.
$$
In particular they do not tend to zero: the subsequence with $n=6k+1$ is always $\sqrt3/2$. The <term test for divergence> therefore shows that \b[the <series> diverges]. Although each block of six sums to zero, that only makes a subsequence of <partial sums> zero; it does not make the full sequence of <partial sums> convergent.
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