# Closed form of $\sum_{n=0}^{\infty} \frac{\cos(\alpha \, n)}{n!}$

How would I find the closed form of

$$f(\alpha) = \sum_{n=0}^{\infty} \frac{\cos(\alpha \, n)}{n!}$$

My old pal Wolfram tells me that

$$f(\alpha) = \mathrm{e}^{\cos(\alpha)}\cos(\sin(\alpha))$$

I've attempted writing out the first couple terms for different values of $\alpha$, but I haven't quite figured out how to arrive at the result.

• Use De Moivre's Theorem $\exp(i \alpha) =\cos \theta + i\sin \theta$ and the look at the real part. – Donald Splutterwit Feb 9 '17 at 0:17

$$\cos(\alpha n)=\frac{1}{2}\left(\left(e^{i\alpha}\right)^n+\left(e^{-i\alpha}\right)^n\right)$$
Alternatively, this can be written as (when $\alpha$ is real) as
$\cos(\alpha n)$ is the real part of $\left(e^{i\alpha}\right)^n$.