Answer
$$cos\frac{\sqrt \pi}{3}$$
Work Step by Step
Given: $$\Sigma_{n=0}^{\infty}\frac{(-1)^{n}\pi^{n}}{3^{2n}(2n)!}$$
As we know, $$cosx=\Sigma_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!}$$
Re-write the given equation as
$$\Sigma_{n=0}^{\infty}\frac{(-1)^{n}\pi^{n}}{3^{2n}(2n)!}=\Sigma_{n=0}^{\infty}\frac{(-1)^{n}(\sqrt \pi/3)^{2n}}{(2n)!}$$
Now we can see that $x=\frac{\sqrt \pi}{3}$, thus, the series sum is
$$\Sigma_{n=0}^{\infty}\frac{(-1)^{n}\pi^{n}}{3^{2n}(2n)!}=cos\frac{\sqrt \pi}{3}$$
