Answer
Diverges
Work Step by Step
Here, we have the given series $\Sigma_{n=1}^{\infty} (\dfrac{(-1)^n 3^{n-2}}{2^2})$
Re-write the above series: $\Sigma_{n=1}^{\infty} (\dfrac{(-1)^n 3^{n-2}}{2^2})=(\dfrac{-3^{-1}}{2})+\dfrac{1}{4}+\dfrac{-3}{8}+.....$
From the above series, we can see that the series shows a geometric series with common ratio $|r|=|\dfrac{-3}{2}| \implies \dfrac{3}{2} \gt 1$.
Hence, we can conclude that the given series diverges.
