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

Proof
Let $\map f x = e^{\cos x}$.

Then:

By definition of Taylor series:


 * $\map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$

and so expanding about $\xi = 0$: