Answer
Absolutely Convergent
Work Step by Step
Consider the Limit Comparison Test for a series $\Sigma a_n$ such that $p_n=(-1)^n q_n$; $q_n \geq 0$ for all $n$.
In order to solve this series, we have:
$q_n=\dfrac{1}{n^2}$ $\lim\limits_{n \to \infty} \dfrac{p_n}{q_n}=\lim\limits_{n \to \infty} \dfrac{n^2}{(2n+1)^2}+ \dfrac{n^2}{(2n+2)^2}\\=\lim\limits_{n \to \infty} \dfrac{1}{(2+\dfrac{1}{n})^2}+ \dfrac{1}{(2+\dfrac{2}{n})^2} \\= \dfrac{1}{2} \ne 0 \ne \infty$
This implies that the series is Absolutely Convergent by the Limit Comparison Test.
