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 $f \left({x}\right) = e^{\sin x}$.

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 $\xi = 0$:

No pattern is immediately apparent.