determine whether the series is absolutely convergent, conditionally convergent, or divergent

Question

does the series $((-2)^n)/(n^2)$ from $n=1$ to infinity converge or diverge? Is the ratio test applied?

Answer 1

$a_n = \dfrac{(-2)^n}{n^2}$. If the series converges, $a_n \to 0$ as $n \to \infty$, thus all subsequences must converge to $0$, and $a_{2n}$ must converge to $0$. But this subsequence diverges to $\infty$, thus the series diverges.

Answer 2

By the ratio test, as $n \to \infty$, $$ \left|\frac{(-2)^{n+1}}{(n+1)^2}\cdot \frac{n^2}{(-2)^{n}} \right|=\left|\frac{2}{\left(1+\frac1n\right)^2} \right|\to 2>1 $$ the given series is divergent.