Power Series Expansion for Exponential of Sine of x

Theorem

 * $e^{\sin x} = 1 + x + \dfrac {x^2} 2 - \dfrac {x^4} 8 - \dfrac {x^5} {15} + \cdots$

for all $x \in \R$.

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

Then:

By definition of Taylor series:


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

and so expanding about $\xi = 0$:

No pattern is immediately apparent.