= Solution
The <alternating series test> says that if $a_n\geq0$ decreases to $0$, then
$$
\sum_{n=1}^{\infty}(-1)^na_n
$$
converges. Taking $a_n=n^{-1/2}$ proves convergence of the given series.
It is not <absolute convergence>, because the series of absolute values is the <p-series>
$$
\sum_{n=1}^{\infty}\frac1{\sqrt n},
$$
which diverges since $p=1/2\leq1$. Thus the original series has <conditional convergence>.
For an explicit divergent <rearrangement of a series>, take unused positive terms, which are the even-indexed terms, until the partial sum exceeds $1$, then take the first unused negative term. Next take positive terms until the sum exceeds $2$, then the next unused negative term, and continue. Both the positive and negative subseries have infinite total magnitude, so this procedure uses every term. The negative term inserted at stage $m$ tends to zero, while the preceding partial sum exceeds $m$; consequently these rearranged partial sums tend to $+\infty$. This is a divergent series with exactly the prescribed terms.
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