A series with an equivalent that converges but does not

Equivalent. $\frac{v_n}{u_n}=\frac{\sqrt n}{\sqrt n+(-1)^n}\to1$.

$\sum u_n$ converges by the alternating series test ($\frac1{\sqrt n}$ decreases to $0$).

Rewriting $v_n$.
$$v_n=\frac{(-1)^n\left(\sqrt n-(-1)^n\right)}{n-1}=\frac{(-1)^n\sqrt n}{n-1}-\frac1{n-1}.$$

First part. $w_n=\frac{\sqrt n}{n-1}$ is positive, tends to $0$, and is eventually decreasing (for $n\ge4$, its derivative in $n$ is negative). So $\sum\frac{(-1)^n\sqrt n}{n-1}$ converges.

Second part. $\sum\frac1{n-1}$ diverges (harmonic).

Conclusion. $\sum v_n$ is a convergent series minus a divergent one, hence diverges.

Moral. For series of non-constant sign, *equivalent terms do not imply the same nature*. The small perturbation $(-1)^n$ in the denominator contributes a divergent harmonic-like piece $-\frac1n$ hidden in the next-order term.