Answer
Diverges
Work Step by Step
Let us consider $a_n=(-1)^{n+1}\dfrac{\sqrt[3]{n+1}}{\sqrt n+1}$
and $u_n$ refers to all positive values of $n$.
Here, we have $f(n)=(-1)^{n+1}\dfrac{\sqrt[3]{n+1}}{\sqrt n+1}$
This implies that $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (-1)^{n+1}\dfrac{\sqrt[3]{n+1}}{\sqrt n+1}=3\ne 0\ne$
Hence, the series diverges by the Nth Term Test for Divergence.