## Calculus 8th Edition

$$cos\frac{\sqrt \pi}{3}$$
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 $$\Sigma_{n=0}^{\infty}\frac{(-1)^{n}\pi^{n}}{3^{2n}(2n)!}=cos\frac{\sqrt \pi}{3}$$