Answer
Conditionally Convergent
Work Step by Step
A series $ \Sigma a_n$ is said to be absolutely convergent when $ \Sigma |a_n|$ is convergent.
We notice that
$ \Sigma_{n=2}^{\infty} |\dfrac{(-1)^n}{2n}|=\dfrac{1}{2} \Sigma_{n=2}^{\infty} \dfrac{1}{n}$
We see a p-series with common ratio $r=1$. When the common ratio $r \gt 1$, then a p-series is convergent.
Therefore, the given series is not Absolutely Convergent .
The Alternating Series Test states:
Consider a series $\Sigma a_n$ such that $A_n=(-1)^n B_n$; $B_n \geq 0$ for all $n$
If the following conditions are satisfied, then the series converges:
a) $\lim\limits_{n \to \infty} B_n=0$;
b) $B_n$ is a decreasing sequence.
We notice that $B_n=\dfrac{1}{2n}$
a) $\lim\limits_{n \to \infty} A_n=\lim\limits_{n \to \infty}\dfrac{1}{2n}=0$
b) We notice that the series is a decreasing sequence.
Therefore, the given series is Conditionally Convergent by the Alternating Series Test.
