Answer
Diverges
Work Step by Step
Consider $u_n=(-1)^{n+1}\dfrac{2^n}{ n^2}$
Here, $u_n$ is positive for all the values of $n$.
Now $\lim\limits_{n \to \infty} u_n=(-1)^{n+1}\dfrac{2^n}{ n^2}=DNE$
The series diverges by the nth Term Test for Divergence.