= Solution
The <alternating series test> states that if $b_n\geq0$ decreases to zero, then $\sum(-1)^nb_n$ converges. Indeed, its even and odd <partial sums> are monotone in opposite directions and bound each other, while their difference tends to zero. Here $b_n=1/(1+\sqrt n)$ decreases to zero, so the <series> converges.
It fails <absolute convergence>, since for $n\geq1$,
$$
\frac1{1+\sqrt n}\geq\frac1{2\sqrt n}.
$$
The <p-series> $\sum n^{-1/2}$ diverges. This also follows by comparison with the divergent <harmonic series>, because $n^{-1/2}\geq n^{-1}$. The <comparison test for series> therefore shows that $\sum b_n$ diverges. Thus
$$
\boxed{\sum_{n\geq1}\frac{(-1)^n}{1+\sqrt n}\text{ converges conditionally}.}
$$
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