Answer
Absolutely Convergent
Work Step by Step
A series $ \Sigma a_n$ is said to be absolutely convergent when $ \Sigma |a_n|$ is convergent.
When the common ratio $r \gt 1$, then a p-series is convergent.
To use the Limit Comparison Test, we consider a series $\Sigma a_n$ such that $A_n=(-1)^n B_n$; $B_n \geq 0$ for all $n$
It has been noticed that $B_n=\dfrac{1}{n^2}$
Now, $\lim\limits_{n \to \infty} \dfrac{A_n}{B_n}=\lim\limits_{n \to \infty} \dfrac{n^2}{(2n+1)^2}+ \dfrac{n^2}{(2n+2)^2}$
or, $=\lim\limits_{n \to \infty} \dfrac{1}{(2+1/n)^2}+ \dfrac{1}{(2+2/n)^2}$
or, $= \dfrac{1}{2} \ne 0 \ne \infty$
Therefore, the given series is Absolutely Convergent by the Limit Comparison Test.
