Find $\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}$

Question

How could one find $$\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}\,?$$

I tried to use Fourier series and integrals depending on a parameter to reduce the problem to a differential equation, but that didn't work.

Answer 1

Using Intuition behind euler's formula

$$\sum_{n=0}^{\infty}{\frac{\cos(nx)}{n!}}=$$

$$=\text{ real part of }\sum_{n=0}^{\infty}\dfrac{(e^{ix})^n}{n!}$$

$$\sum_{n=0}^{\infty}\dfrac{(e^{ix})^n}{n!}=e^{\cos x+i\sin x}=e^{\cos x}(\cos(\sin x)+i\sin(\sin x))$$

Answer 2

\begin{align*} \sum_{n=0}^{+\infty}\frac{\cos(nx)}{n!} =&\Re\left[\sum_{n=0}^{+\infty}\frac{(e^{ix})^n}{n!}\right]\\ =&\Re\left[e^{e^{ix}}\right]\\ =&\Re[e^{\cos x+i\sin x}]\\ =& e^{\cos x}\cdot\cos(\sin x) \end{align*}