Answer
Convergent
Work Step by Step
$\lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
= \lim\limits_{n \to \infty} \left|
(-1)^{(n+1)} \frac{\pi^{2n+1}}{(2n+1)!}
(-1)^{-n} \frac{(2n)!} {\pi^{2n}}
\right|
\\= \lim\limits_{n \to \infty}
\frac{\pi }{(2n+1)}
=0
$
The series is absolutely convergent, as a result is convergent.