Is the value of $\sum_{x=0}^{\infty}\frac{\cos(\pi x)}{x!}=1/e$?

Question

What is the value of $$\sum_{x=0}^{\infty}\frac{\cos (\pi x)}{x!}$$ I wrote $\cos (\pi x)=R (e^{ix}).e^{\pi} $ but the $x! $ is a trouble can someone help me out. And if value isnt $1/e $ then can we get a closed form.Thanks

Answer 1

As $e^y=\sum_{r=0}^\infty\dfrac{y^r}{r!}$

$$\sum_{x=0}^\infty\dfrac{e^{i\pi x}}{x!}=\sum_{x=0}^\infty\dfrac{(e^{i\pi})^x}{x!}=e^{(e^{i\pi})}$$

Now $e^{i\pi}=\cos\pi+i\sin\pi=?$

Answer 2

Probably you mean $x=0$ to $\infty$, and so $x$ is an integer. If so, $\cos(\pi x) = (-1)^x$. Then your sum equals $$\sum_{x=0}^{\infty} \frac{(-1)^x}{x!},$$ which is the Taylor series for $e^x$ evaluated at $-1$. So it equals $1/e$.