Power Series Expansion for Exponential of Cosine of x/Proof 1

Proof
Let $f \left({x}\right) = e^{\cos x}$.

Then:

By definition of Taylor series:


 * $f \left({x}\right) \sim \displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$

and so expanding about $\xi = 0$: