Answer
Absolutely convergent
Work Step by Step
Suppose that $a_n=\frac{(-1)^n(n+1)3^n}{2^{2n+1}}$.
It can be checked that $\frac{a_{n+1}}{a_n}=-\frac{3(n+2)}{4(n+1)}$.
Then,
$\lim |\frac{a_{n+1}}{a_n}|=\lim |-\frac{3(n+2)}{4(n+1)}|=\lim |-\frac{3(1+2/n)}{4(1+1/n)}|=|-\frac{3(1+0)}{4(1+0)}|=\frac{3}{4}<1$
Using The Ratio Test, the series is absolutely convergent.
