Answer
Diverges
Work Step by Step
Let us consider $a_n=(-1)^{n+1}\dfrac{2^{(n)}}{ n^{(2)}}$ and $u_n$ refers to all positive values of $n$.
Here, $\lim\limits_{n \to \infty} a_n=(-1)^{n+1}\dfrac{2^n/ n^2}{1}=(-1) (\infty)=\infty$
Hence, the series diverges by the Nth Term Test for Divergence.